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<title>Complexification (Lie group)</title>

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Complexification (Lie group)

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Lie_group" title="Lie group">Lie groups</a> and <a href="Lie_algebra" title="Lie algebra">Lie algebras</a></th></tr><tr><td class="sidebar-image skin-invert-image" style="padding-bottom:0.9em;"></td></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Classical_group" title="Classical group">Classical groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist">

<ul><li><a href="General_linear_group" title="General linear group">General linear</a> GL(n)</li>

<li><a href="Special_linear_group" title="Special linear group">Special linear</a> SL(n)</li>

<li><a href="Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(n)</li>

<li><a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(n)</li>

<li><a href="Unitary_group" title="Unitary group">Unitary</a> U(n)</li>

<li><a href="Special_unitary_group" title="Special unitary group">Special unitary</a> SU(n)</li>

<li><a href="Symplectic_group" title="Symplectic group">Symplectic</a> Sp(n)</li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Simple_Lie_group" title="Simple Lie group">Simple Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks hlist" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;">

Classical</th></tr><tr><td class="sidebar-content">

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Exceptional</th></tr><tr><td class="sidebar-content">

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Table_of_Lie_groups" title="Table of Lie groups">Other Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">

<ul><li><a href="Circle_group" title="Circle group">Circle</a></li>

<li><a href="Lorentz_group" title="Lorentz group">Lorentz</a></li>

<li><a href="Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li>

<li><a href="Conformal_group" title="Conformal group">Conformal group</a></li>

<li><a href="Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li>

<li><a href="Euclidean_group" title="Euclidean group">Euclidean</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Lie_algebra" title="Lie algebra">Lie algebras</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist">

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie group–Lie algebra correspondence</a></li>

<li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">Exponential map</a></li>

<li><a href="Adjoint_representation" title="Adjoint representation">Adjoint representation</a></li>

<li><div class="hlist"><ul><li><a href="Killing_form" title="Killing form">Killing form</a></li><li><a href="Index_of_a_Lie_algebra" title="Index of a Lie algebra">Index</a></li></ul></div></li>

<li><a href="Simple_Lie_algebra" title="Simple Lie algebra">Simple Lie algebra</a></li>

<li><a href="Loop_algebra" title="Loop algebra">Loop algebra</a></li>

<li><a href="Affine_Lie_algebra" title="Affine Lie algebra">Affine Lie algebra</a></li></ul>

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<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Semisimple_Lie_algebra" title="Semisimple Lie algebra">Semisimple Lie algebra</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist">

<ul><li><a href="Dynkin_diagram" title="Dynkin diagram">Dynkin diagrams</a></li>

<li><a href="Cartan_subalgebra" title="Cartan subalgebra">Cartan subalgebra</a></li>

<li><div class="hlist"><ul><li><a href="Root_system" title="Root system">Root system</a></li><li><a href="Weyl_group" title="Weyl group">Weyl group</a></li></ul></div></li>

<li><a href="Split_Lie_algebra" title="Split Lie algebra">Split Lie algebra</a></li>

<li><a href="Compact_Lie_algebra" title="Compact Lie algebra">Compact Lie algebra</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Representation_theory" title="Representation theory">Representation theory</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist">

<ul><li><a href="Representation_of_a_Lie_group" title="Representation of a Lie group">Lie group representation</a></li>

<li><a href="Lie_algebra_representation" title="Lie algebra representation">Lie algebra representation</a></li>

<li><a href="Representation_theory_of_semisimple_Lie_algebras" title="Representation theory of semisimple Lie algebras">Representation theory of semisimple Lie algebras</a></li>

<li><a href="Representations_of_classical_Lie_groups" title="Representations of classical Lie groups">Representations of classical Lie groups</a></li>

<li><a href="Theorem_of_the_highest_weight" title="Theorem of the highest weight">Theorem of the highest weight</a></li>

<li><a href="Borel%E2%80%93Weil%E2%80%93Bott_theorem" title="Borel–Weil–Bott theorem">Borel–Weil–Bott theorem</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Lie groups in <a href="Physics" title="Physics">physics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist">

<ul><li><a href="Particle_physics_and_representation_theory" title="Particle physics and representation theory">Particle physics and representation theory</a></li>

<li><a href="Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">Lorentz group representations</a></li>

<li><a href="Representation_theory_of_the_Poincar%C3%A9_group" title="Representation theory of the Poincaré group">Poincaré group representations</a></li>

<li><a href="Representation_theory_of_the_Galilean_group" title="Representation theory of the Galilean group">Galilean group representations</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">

<ul><li><a href="Sophus_Lie" title="Sophus Lie">Sophus Lie</a></li>

<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li>

<li><a href="Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a></li>

<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li>

<li><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li>

<li><a href="Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a></li>

<li><a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a></li>

<li><a href="Armand_Borel" title="Armand Borel">Armand Borel</a></li></ul>

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</tr><tr><td class="sidebar-below plainlist">

<ul><li><a href="Glossary_of_Lie_groups_and_Lie_algebras" title="Glossary of Lie groups and Lie algebras">Glossary</a></li>

<li><a href="Table_of_Lie_groups" title="Table of Lie groups">Table of Lie groups</a></li></ul></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the complexification or universal complexification of a <a href="Real_Lie_group" class="mw-redirect" title="Real Lie group">real Lie group</a> is given by a continuous homomorphism of the group into a <a href="Complex_Lie_group" title="Complex Lie group">complex Lie group</a> with the <a href="Universal_property" title="Universal property">universal property</a> that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a <a href="Complex_analytic_function" class="mw-redirect" title="Complex analytic function">complex analytic</a> homomorphism between the complex Lie groups. The complexification, which always exists, is unique <a href="Natural_isomorphism" class="mw-redirect" title="Natural isomorphism">up to unique isomorphism</a>. Its <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> is a quotient of the <a href="Complexification" title="Complexification">complexification</a> of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

For <a href="Compact_Lie_group" class="mw-redirect" title="Compact Lie group">compact Lie groups</a>, the complexification, sometimes called the Chevalley complexification after <a href="Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a>, can be defined as the group of complex characters of the <a href="Hopf_algebra" title="Hopf algebra">Hopf algebra</a> of <a href="Representative_function" class="mw-redirect" title="Representative function">representative functions</a>, i.e. the <a href="Matrix_coefficient" title="Matrix coefficient">matrix coefficients</a> of finite-dimensional <a href="Unitary_representation" title="Unitary representation">representations</a> of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex <a href="General_linear_group" title="General linear group">general linear group</a>. It consists of operators with <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> g = u • exp iX, where u is a unitary operator in the compact group and X is a <a href="Skew-adjoint_operator" class="mw-redirect" title="Skew-adjoint operator">skew-adjoint operator</a> in its Lie algebra. In this case the complexification is a <a href="Algebraic_group" title="Algebraic group">complex algebraic group</a> and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.

<div class="mw-heading mw-heading2"><h2 id="Universal_complexification">Universal complexification</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

If G is a Lie group, a universal complexification is given by a complex Lie group GC and a continuous homomorphism φ: G → GC with the universal property that, if f: G → H is an arbitrary continuous homomorphism into a complex Lie group H, then there is a unique complex analytic homomorphism F: GC → H such that f = F ∘ φ.

Universal complexifications always exist and are unique up to a unique complex analytic isomorphism (preserving inclusion of the original group).

<div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3></div>

If G is connected with Lie algebra 𝖌, then its <a href="Universal_covering_group" class="mw-redirect" title="Universal covering group">universal covering group</a> G is simply connected. Let GC be the simply connected complex Lie group with Lie algebra 𝖌C = 𝖌 ⊗ C, let Φ: G → GC be the natural homomorphism (the unique morphism such that Φ*: 𝖌 ↪ 𝖌 ⊗ C is the canonical inclusion) and suppose π: G → G is the universal covering map, so that ker π is the fundamental group of G. We have the inclusion Φ(ker π) ⊂ Z(GC), which follows from the fact that the kernel of the adjoint representation of GC equals its centre, combined with the equality

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<annotation encoding="application/x-tex">{\displaystyle (C_{\Phi (k)})_{*}\circ \Phi _{*}=\Phi _{*}\circ (C_{k})_{*}=\Phi _{*}}</annotation>

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</math><img src="./6bc5171bd8a9dd118ea72a77e2e0e0e8297c3453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.477ex; height:3.176ex;" alt="{\displaystyle (C_{\Phi (k)})_{*}\circ \Phi _{*}=\Phi _{*}\circ (C_{k})_{*}=\Phi _{*}}" loading="lazy"></dd></dl>

which holds for any k ∈ ker π. Denoting by Φ(ker π)* the smallest closed normal Lie subgroup of GC that contains Φ(ker π), we must now also have the inclusion Φ(ker π)* ⊂ Z(GC). We define the universal complexification of G as

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathbf {C} }={\frac {\mathbf {G} _{\mathbf {C} }}{\Phi (\ker \pi )^{*}}}.}</annotation>

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</math><img src="./a64912bfcbf9776ba0fa250fe61a15d88fc74249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.438ex; height:6.176ex;" alt="{\displaystyle G_{\mathbf {C} }={\frac {\mathbf {G} _{\mathbf {C} }}{\Phi (\ker \pi )^{*}}}.}" loading="lazy"></dd></dl>

In particular, if G is simply connected, its universal complexification is just GC.<a href="#cite_note-1">[1]</a>

The map φ: G → GC is obtained by passing to the quotient. Since π is a surjective submersion, smoothness of the map πC ∘ Φ implies smoothness of φ.

For non-connected Lie groups G with identity component Go and component group Γ = G / Go, the extension

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<annotation encoding="application/x-tex">{\displaystyle \{1\}\rightarrow G^{o}\rightarrow G\rightarrow \Gamma \rightarrow \{1\}}</annotation>

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</math><img src="./57b0d8faf186047b444971259f1852e9b25a5135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.567ex; height:2.843ex;" alt="{\displaystyle \{1\}\rightarrow G^{o}\rightarrow G\rightarrow \Gamma \rightarrow \{1\}}" loading="lazy"></dd></dl>

induces an extension

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<annotation encoding="application/x-tex">{\displaystyle \{1\}\rightarrow (G^{o})_{\mathbf {C} }\rightarrow G_{\mathbf {C} }\rightarrow \Gamma \rightarrow \{1\}}</annotation>

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</math><img src="./3bd1059f4569bbd209023e9f90e48089a820ac81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.572ex; height:2.843ex;" alt="{\displaystyle \{1\}\rightarrow (G^{o})_{\mathbf {C} }\rightarrow G_{\mathbf {C} }\rightarrow \Gamma \rightarrow \{1\}}" loading="lazy"></dd></dl>

and the complex Lie group GC is a complexification of G.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading4"><h4 id="Proof_of_the_universal_property">Proof of the universal property</h4></div>

The map φ: G → GC indeed possesses the universal property which appears in the above definition of complexification. The proof of this statement naturally follows from considering the following instructive diagram.

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon G\rightarrow H}">

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</math><img src="./9164e9702cb24cb7d146197f10db5efee96c9b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.817ex; height:2.509ex;" alt="{\displaystyle f\colon G\rightarrow H}" loading="lazy"> is an arbitrary smooth homomorphism of Lie groups with a complex Lie group as the codomain.

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</style><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="color:green;background:lightgrey;">Existence of the map F</div><div class="hidden-content mw-collapsible-content" style="">

For simplicity, we assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is connected. To establish the existence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, we first naturally extend the morphism of Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}\colon {\mathfrak {g}}\rightarrow {\mathfrak {h}}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{*}\colon {\mathfrak {g}}\rightarrow {\mathfrak {h}}}</annotation>

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</math><img src="./b9e3e1aa2f1fe0451bdd8e5be6f805e371e79519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.224ex; height:2.509ex;" alt="{\displaystyle f_{*}\colon {\mathfrak {g}}\rightarrow {\mathfrak {h}}}" loading="lazy"> to the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}_{*}\colon {\mathfrak {g}}_{\mathbf {C} }\rightarrow {\mathfrak {h}}}">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {f}}_{*}\colon {\mathfrak {g}}_{\mathbf {C} }\rightarrow {\mathfrak {h}}}</annotation>

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</math><img src="./7e3c35ee934bb7583addef91b79b170c52e3dcc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.167ex; height:3.509ex;" alt="{\displaystyle {\overline {f}}_{*}\colon {\mathfrak {g}}_{\mathbf {C} }\rightarrow {\mathfrak {h}}}" loading="lazy"> of complex Lie algebras. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{\mathbf {C} }}</annotation>

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</math><img src="./0efae18a779f6ccdc0f1843010aa531f7f305149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.699ex; height:2.509ex;" alt="{\displaystyle \mathbf {G} _{\mathbf {C} }}" loading="lazy"> is simply connected, Lie's second fundamental theorem now provides us with a unique complex analytic morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}\colon \mathbf {G} _{\mathbf {C} }\rightarrow H}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {F}}\colon \mathbf {G} _{\mathbf {C} }\rightarrow H}</annotation>

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</math><img src="./3f46588ca3ddc5622b59ace229ce7d1a04076080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.426ex; height:3.343ex;" alt="{\displaystyle {\overline {F}}\colon \mathbf {G} _{\mathbf {C} }\rightarrow H}" loading="lazy"> between complex Lie groups, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\overline {F}})_{*}={\overline {f}}_{*}}">

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</mover>

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<annotation encoding="application/x-tex">{\displaystyle ({\overline {F}})_{*}={\overline {f}}_{*}}</annotation>

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</math><img src="./f49ba10ee0aa9731a6b359e46daf9ebde82b99cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.516ex; height:3.509ex;" alt="{\displaystyle ({\overline {F}})_{*}={\overline {f}}_{*}}" loading="lazy">. We define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon G_{\mathbf {C} }\rightarrow H}">

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<annotation encoding="application/x-tex">{\displaystyle F\colon G_{\mathbf {C} }\rightarrow H}</annotation>

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</math><img src="./672071efc33d471aca3f66147723475670531e88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.877ex; height:2.509ex;" alt="{\displaystyle F\colon G_{\mathbf {C} }\rightarrow H}" loading="lazy"> as the map induced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {F}}}</annotation>

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</math><img src="./d122dfa2be8a341b1c30e7b3405af6ac2c157105.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.016ex; height:3.009ex;" alt="{\displaystyle {\overline {F}}}" loading="lazy">, that is: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(g\,\Phi (\ker \pi )^{*})={\overline {F}}(g)}">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle F(g\,\Phi (\ker \pi )^{*})={\overline {F}}(g)}</annotation>

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</math><img src="./f009d89ee2555d2a4230ddce26fe63422398e0bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.525ex; height:3.509ex;" alt="{\displaystyle F(g\,\Phi (\ker \pi )^{*})={\overline {F}}(g)}" loading="lazy"> for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in \mathbf {G} _{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle g\in \mathbf {G} _{\mathbf {C} }}</annotation>

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</math><img src="./ae5da46a0c8b554e436496c6dd89f7fe64c9af76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.655ex; height:2.509ex;" alt="{\displaystyle g\in \mathbf {G} _{\mathbf {C} }}" loading="lazy">. To show well-definedness of this map (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\ker \pi )^{*}\subset \ker {\overline {F}}}">

<msup>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\ker \pi )^{*}\subset \ker {\overline {F}}}</annotation>

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</math><img src="./be9613233e8c3da42c423eadc4ecafba30f43a7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.105ex; height:3.509ex;" alt="{\displaystyle \Phi (\ker \pi )^{*}\subset \ker {\overline {F}}}" loading="lazy">), consider the derivative of the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}\circ \Phi }">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {F}}\circ \Phi }</annotation>

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</math><img src="./4cfa6be6d8e494775039a813dd8ea3f2d07fffdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.889ex; height:3.009ex;" alt="{\displaystyle {\overline {F}}\circ \Phi }" loading="lazy">. For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in T_{e}\mathbf {G} \cong {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle v\in T_{e}\mathbf {G} \cong {\mathfrak {g}}}</annotation>

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</math><img src="./166ddb3b888e88ad33ddda8ee996af45de6b2340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.695ex; height:2.509ex;" alt="{\displaystyle v\in T_{e}\mathbf {G} \cong {\mathfrak {g}}}" loading="lazy">, we have

<mover>

</mover>

</mrow>

<msub>

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</msub>

<msub>

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</msub>

<mover>

</mover>

</mrow>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle ({\overline {F}})_{*}\Phi _{*}v=({\overline {F}})_{*}(v\otimes 1)=f_{*}\pi _{*}v}</annotation>

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</math><img src="./3819d409daa72baea01e52e4f64d336f02cc4e82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.456ex; height:3.509ex;" alt="{\displaystyle ({\overline {F}})_{*}\Phi _{*}v=({\overline {F}})_{*}(v\otimes 1)=f_{*}\pi _{*}v}" loading="lazy">,</dd></dl>

which (by simple connectedness of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} }</annotation>

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</math><img src="./f6d9c60d3cf462a9812e9a9d021d17c7bc272a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.101ex; height:2.176ex;" alt="{\displaystyle \mathbf {G} }" loading="lazy">) implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}\circ \Phi =f\circ \pi }">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {F}}\circ \Phi =f\circ \pi }</annotation>

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</math><img src="./77fff8d9c7d4ed71cf2af73c14977f4080bc16a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.792ex; height:3.343ex;" alt="{\displaystyle {\overline {F}}\circ \Phi =f\circ \pi }" loading="lazy">. This equality finally implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\ker \pi )\subset \ker {\overline {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\ker \pi )\subset \ker {\overline {F}}}</annotation>

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</math><img src="./d98a017b289b0a32f64dbbc713404d34959fde23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.051ex; height:3.509ex;" alt="{\displaystyle \Phi (\ker \pi )\subset \ker {\overline {F}}}" loading="lazy">, and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker {\overline {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle \ker {\overline {F}}}</annotation>

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</math><img src="./ba45f6014036834fc80b8c485031405d76ccc67b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.574ex; height:3.009ex;" alt="{\displaystyle \ker {\overline {F}}}" loading="lazy"> is a closed normal Lie subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {G} _{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{\mathbf {C} }}</annotation>

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</math><img src="./0efae18a779f6ccdc0f1843010aa531f7f305149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.699ex; height:2.509ex;" alt="{\displaystyle \mathbf {G} _{\mathbf {C} }}" loading="lazy">, we also have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\ker \pi )^{*}\subset \ker {\overline {F}}}">

<msup>

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<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (\ker \pi )^{*}\subset \ker {\overline {F}}}</annotation>

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</math><img src="./be9613233e8c3da42c423eadc4ecafba30f43a7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.105ex; height:3.509ex;" alt="{\displaystyle \Phi (\ker \pi )^{*}\subset \ker {\overline {F}}}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{\mathbb {C} }}</annotation>

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</math><img src="./c3d7ede1eef46a60e738e1e57219361a4d47b456.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.744ex; height:2.009ex;" alt="{\displaystyle \pi _{\mathbb {C} }}" loading="lazy"> is a complex analytic surjective submersion, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is complex analytic since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {F}}}</annotation>

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</math><img src="./d122dfa2be8a341b1c30e7b3405af6ac2c157105.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.016ex; height:3.009ex;" alt="{\displaystyle {\overline {F}}}" loading="lazy"> is. The desired equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\circ \varphi =f}">

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<annotation encoding="application/x-tex">{\displaystyle F\circ \varphi =f}</annotation>

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</math><img src="./1a2191ab92f4cc7eb2203d41f12f7c6c0d0d1f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.833ex; height:2.676ex;" alt="{\displaystyle F\circ \varphi =f}" loading="lazy"> is imminent.

</div></div>

<div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="color:green;background:lightgrey;">Uniqueness of the map F</div><div class="hidden-content mw-collapsible-content" style="">

To show uniqueness of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1},F_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{1},F_{2}}</annotation>

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</math><img src="./ddbaefad7d000285a069031a623cdc454e4b79e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.132ex; height:2.509ex;" alt="{\displaystyle F_{1},F_{2}}" loading="lazy"> are two maps with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}\circ \varphi =F_{2}\circ \varphi =f}">

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<annotation encoding="application/x-tex">{\displaystyle F_{1}\circ \varphi =F_{2}\circ \varphi =f}</annotation>

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</math><img src="./cd56e0ece38f6a5e4c62f7fefd6ef6669e0d6c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.003ex; height:2.676ex;" alt="{\displaystyle F_{1}\circ \varphi =F_{2}\circ \varphi =f}" loading="lazy">. Composing with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> from the right and differentiating, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{1})_{*}(\pi _{\mathbf {C} })_{*}\Phi _{*}=(F_{2})_{*}(\pi _{\mathbf {C} })_{*}\Phi _{*}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (F_{1})_{*}(\pi _{\mathbf {C} })_{*}\Phi _{*}=(F_{2})_{*}(\pi _{\mathbf {C} })_{*}\Phi _{*}}</annotation>

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</math><img src="./bdfba7f5d0bddb2fc6d1e30fe2b90e908ebd94c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.961ex; height:2.843ex;" alt="{\displaystyle (F_{1})_{*}(\pi _{\mathbf {C} })_{*}\Phi _{*}=(F_{2})_{*}(\pi _{\mathbf {C} })_{*}\Phi _{*}}" loading="lazy">, and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{*}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{*}}</annotation>

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</math><img src="./b3eb90c18cf6cecb4daf60472ea9ad4bffede2f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Phi _{*}}" loading="lazy"> is the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}\hookrightarrow {\mathfrak {g}}_{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}\hookrightarrow {\mathfrak {g}}_{\mathbf {C} }}</annotation>

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</math><img src="./8ed29b4e7cc4de14f4a2ffd6b8303f6581178838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.848ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {g}}\hookrightarrow {\mathfrak {g}}_{\mathbf {C} }}" loading="lazy">, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{1})_{*}(\pi _{\mathbf {C} })_{*}=(F_{2})_{*}(\pi _{\mathbf {C} })_{*}}">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle (F_{1})_{*}(\pi _{\mathbf {C} })_{*}=(F_{2})_{*}(\pi _{\mathbf {C} })_{*}}</annotation>

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</math><img src="./6af7d97dcd398e81edabf4add167f7490e456daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.496ex; height:2.843ex;" alt="{\displaystyle (F_{1})_{*}(\pi _{\mathbf {C} })_{*}=(F_{2})_{*}(\pi _{\mathbf {C} })_{*}}" loading="lazy">. But <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{\mathbf {C} }}</annotation>

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</math><img src="./be0878abf0d0f4f06a8ec93860993171f54f436d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.923ex; height:2.009ex;" alt="{\displaystyle \pi _{\mathbf {C} }}" loading="lazy"> is a submersion, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{1})_{*}=(F_{2})_{*}}">

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<annotation encoding="application/x-tex">{\displaystyle (F_{1})_{*}=(F_{2})_{*}}</annotation>

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</math><img src="./140892c83ec9fb1b453ea9e514a177bc8e4f723b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.923ex; height:2.843ex;" alt="{\displaystyle (F_{1})_{*}=(F_{2})_{*}}" loading="lazy">, thus connectedness of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=F_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{1}=F_{2}}</annotation>

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</math><img src="./4a1387d67417e45de53b554e8b775be7ad3cb236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.196ex; height:2.509ex;" alt="{\displaystyle F_{1}=F_{2}}" loading="lazy">.

</div></div>

<div class="mw-heading mw-heading3"><h3 id="Uniqueness">Uniqueness</h3></div>

The universal property implies that the universal complexification is unique up to complex analytic isomorphism.

<div class="mw-heading mw-heading3"><h3 id="Injectivity">Injectivity</h3></div>

If the original group is linear, so too is the universal complexification and the homomorphism between the two is an inclusion.<a href="#cite_note-3">[3]</a> <a href="#CITEREFOnishchikVinberg1994">Onishchik & Vinberg (1994)</a> give an example of a connected real Lie group for which the homomorphism is not injective even at the Lie algebra level: they take the product of T by the <a href="Universal_covering_group" class="mw-redirect" title="Universal covering group">universal covering group</a> of SL(2,R) and quotient out by the discrete cyclic subgroup generated by an irrational rotation in the first factor and a generator of the center in the second.

<div class="mw-heading mw-heading3"><h3 id="Basic_examples">Basic examples</h3></div>

The following isomorphisms of complexifications of Lie groups with known Lie groups can be constructed directly from the general construction of the complexification.

<ul><li>The complexification of the <a href="Special_unitary_group" title="Special unitary group">special unitary group</a> of 2x2 matrices is</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)_{\mathbf {C} }\cong \mathrm {SL} (2,\mathbf {C} )}</annotation>

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</math><img src="./48711bf9bc6874235d3081c8e494f4001469549d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.386ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (2)_{\mathbf {C} }\cong \mathrm {SL} (2,\mathbf {C} )}" loading="lazy">.</dd></dl></dd></dl>

<dl><dd>This follows from the isomorphism of Lie algebras

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )}</annotation>

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</math><img src="./0cc20b8c4775e10101a6bddc81b2f4276561f19e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:17.559ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )}" loading="lazy">,</dd></dl></dd>

<dd>together with the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (2)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)}</annotation>

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</math><img src="./4478ef936fe905a135afac8386be77964b8bb448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (2)}" loading="lazy"> is simply connected.</dd></dl>

<ul><li>The complexification of the <a href="Special_linear_group" title="Special linear group">special linear group</a> of 2x2 matrices is</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbf {C} )_{\mathbf {C} }\cong \mathrm {SL} (2,\mathbf {C} )\times \mathrm {SL} (2,\mathbf {C} )}</annotation>

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</math><img src="./371831f2acbc0d9fb426ad132cc358bfe8ac4709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.583ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbf {C} )_{\mathbf {C} }\cong \mathrm {SL} (2,\mathbf {C} )\times \mathrm {SL} (2,\mathbf {C} )}" loading="lazy">.</dd></dl></dd></dl>

<dl><dd>This follows from the isomorphism of Lie algebras

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )\oplus {\mathfrak {sl}}(2,\mathbf {C} )}</annotation>

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</math><img src="./de8ac3a28d0a433d89137ad84a606ee66b90a4ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:30.433ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )\oplus {\mathfrak {sl}}(2,\mathbf {C} )}" loading="lazy">,</dd></dl></dd>

<dd>together with the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (2,\mathbf {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbf {C} )}</annotation>

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</math><img src="./605f640f132b2fdc8c73addf4e22ebdf7e682b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.682ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbf {C} )}" loading="lazy"> is simply connected.</dd></dl>

<ul><li>The complexification of the <a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> of 3x3 matrices is</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (3)_{\mathbf {C} }\cong {\frac {\mathrm {SL} (2,\mathbf {C} )}{\mathbf {Z} _{2}}}\cong \mathrm {SO} ^{+}(1,3)}</annotation>

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</math><img src="./cfc8f8f0a2802e54d403834c9df7dce37f11ec22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.165ex; height:6.009ex;" alt="{\displaystyle \mathrm {SO} (3)_{\mathbf {C} }\cong {\frac {\mathrm {SL} (2,\mathbf {C} )}{\mathbf {Z} _{2}}}\cong \mathrm {SO} ^{+}(1,3)}" loading="lazy">,</dd></dl></dd></dl>

<dl><dd>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} ^{+}(1,3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} ^{+}(1,3)}</annotation>

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</math><img src="./21b4aeb08fce7aeafd5c03d81e8fd78204fb4628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.78ex; height:3.009ex;" alt="{\displaystyle \mathrm {SO} ^{+}(1,3)}" loading="lazy"> denotes the proper orthochronous <a href="Lorentz_group" title="Lorentz group">Lorentz group</a>. This follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (2)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)}</annotation>

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</math><img src="./4478ef936fe905a135afac8386be77964b8bb448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (2)}" loading="lazy"> is the universal (double) cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (3)}</annotation>

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</math><img src="./8366fc6e92660ba077b87b745b305a4176b1d1ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (3)}" loading="lazy">, hence:

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)_{\mathbf {C} }\cong {\mathfrak {su}}(2)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )}</annotation>

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</math><img src="./9a25b7fad66b22ca6adab47c4ff2617c1bd49e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:27.394ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)_{\mathbf {C} }\cong {\mathfrak {su}}(2)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )}" loading="lazy">.</dd></dl></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbf {C} )}</annotation>

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</math><img src="./605f640f132b2fdc8c73addf4e22ebdf7e682b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.682ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbf {C} )}" loading="lazy"> is the universal (double) cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} ^{+}(1,3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} ^{+}(1,3)}</annotation>

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</math><img src="./21b4aeb08fce7aeafd5c03d81e8fd78204fb4628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.78ex; height:3.009ex;" alt="{\displaystyle \mathrm {SO} ^{+}(1,3)}" loading="lazy">.</dd></dl>

<ul><li>The complexification of the proper orthochronous Lorentz group is</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} ^{+}(1,3)_{\mathbf {C} }\cong {\frac {\mathrm {SL} (2,\mathbf {C} )\times \mathrm {SL} (2,\mathbf {C} )}{\mathbf {Z} _{2}}}}</annotation>

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</math><img src="./4ca96e35ea25aa956a92b409a32bc590fd2931cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.517ex; height:6.009ex;" alt="{\displaystyle \mathrm {SO} ^{+}(1,3)_{\mathbf {C} }\cong {\frac {\mathrm {SL} (2,\mathbf {C} )\times \mathrm {SL} (2,\mathbf {C} )}{\mathbf {Z} _{2}}}}" loading="lazy">.</dd></dl></dd></dl>

<dl><dd>This follows from the same isomorphism of Lie algebras as in the second example, again using the universal (double) cover of the proper orthochronous Lorentz group.</dd></dl>

<ul><li>The complexification of the special orthogonal group of 4x4 matrices is</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (4)_{\mathbf {C} }\cong {\frac {\mathrm {SL} (2,\mathbf {C} )\times \mathrm {SL} (2,\mathbf {C} )}{\mathbf {Z} _{2}}}}</annotation>

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</math><img src="./433fa7518d79be93fc9ff49a3734eff5d0d7d537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.81ex; height:6.009ex;" alt="{\displaystyle \mathrm {SO} (4)_{\mathbf {C} }\cong {\frac {\mathrm {SL} (2,\mathbf {C} )\times \mathrm {SL} (2,\mathbf {C} )}{\mathbf {Z} _{2}}}}" loading="lazy">.</dd></dl></dd></dl>

<dl><dd>This follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (2)\times \mathrm {SU} (2)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)\times \mathrm {SU} (2)}</annotation>

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</math><img src="./b08d7e4f4ad3e43aba05110c171a0fee91c971f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.855ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (2)\times \mathrm {SU} (2)}" loading="lazy"> is the universal (double) cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (4)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (4)}</annotation>

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</math><img src="./9c67ecc8b7d4d74abfb99e1f735189892c794c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (4)}" loading="lazy">, hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(4)\cong {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(4)\cong {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}</annotation>

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</math><img src="./4099aef6d9dd65552cdeb8525c79487abfea545a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:21.526ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(4)\cong {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}" loading="lazy"> and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(4)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )\oplus {\mathfrak {sl}}(2,\mathbf {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(4)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )\oplus {\mathfrak {sl}}(2,\mathbf {C} )}</annotation>

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</math><img src="./1271d911eec92ae791cf185d26fa363747164ea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:27.953ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(4)_{\mathbf {C} }\cong {\mathfrak {sl}}(2,\mathbf {C} )\oplus {\mathfrak {sl}}(2,\mathbf {C} )}" loading="lazy">.</dd></dl>

The last two examples show that Lie groups with isomorphic complexifications may not be isomorphic. Furthermore, the complexifications of Lie groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (2)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)}</annotation>

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</math><img src="./4478ef936fe905a135afac8386be77964b8bb448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (2)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (2,\mathbf {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbf {C} )}</annotation>

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</math><img src="./605f640f132b2fdc8c73addf4e22ebdf7e682b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.682ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbf {C} )}" loading="lazy"> show that complexification is not an idempotent operation, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{\mathbf {C} })_{\mathbf {C} }\not \cong G_{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle (G_{\mathbf {C} })_{\mathbf {C} }\not \cong G_{\mathbf {C} }}</annotation>

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</math><img src="./63a46947775363bc233fcca6a503b26ba2d2d6db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.355ex; height:2.843ex;" alt="{\displaystyle (G_{\mathbf {C} })_{\mathbf {C} }\not \cong G_{\mathbf {C} }}" loading="lazy"> (this is also shown by complexifications of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (3)}</annotation>

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</math><img src="./8366fc6e92660ba077b87b745b305a4176b1d1ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (3)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} ^{+}(1,3)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} ^{+}(1,3)}</annotation>

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</math><img src="./21b4aeb08fce7aeafd5c03d81e8fd78204fb4628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.78ex; height:3.009ex;" alt="{\displaystyle \mathrm {SO} ^{+}(1,3)}" loading="lazy">).

<div class="mw-heading mw-heading2"><h2 id="Chevalley_complexification">Chevalley complexification</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Hopf_algebra_of_matrix_coefficients">Hopf algebra of matrix coefficients</h3></div>

If G is a compact Lie group, the *-algebra A of matrix coefficients of finite-dimensional unitary representations is a uniformly dense *-subalgebra of C(G), the *-algebra of complex-valued continuous functions on G. It is naturally a <a href="Hopf_algebra" title="Hopf algebra">Hopf algebra</a> with <a href="Comultiplication" class="mw-redirect" title="Comultiplication">comultiplication</a> given by

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\Delta f(g,h)=f(gh).}}</annotation>

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</math><img src="./f6bd2506451e639094a6508d6fd153bfe6aa64b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.801ex; height:2.843ex;" alt="{\displaystyle \displaystyle {\Delta f(g,h)=f(gh).}}" loading="lazy"></dd></dl>

The characters of A are the *-homomorphisms of A into C. They can be identified with the point evaluations f ↦ f(g) for g in G and the comultiplication allows the group structure on G to be recovered. The homomorphisms of A into C also form a group. It is a complex Lie group and can be identified with the complexification GC of G. The *-algebra A is generated by the matrix coefficients of any faithful representation σ of G. It follows that σ defines a faithful complex analytic representation of GC.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading3"><h3 id="Invariant_theory">Invariant theory</h3></div>

The original approach of <a href="#CITEREFChevalley1946">Chevalley (1946)</a> to the complexification of a compact Lie group can be concisely stated within the language of classical <a href="Invariant_theory" title="Invariant theory">invariant theory</a>, described in <a href="#CITEREFWeyl1946">Weyl (1946)</a>. Let G be a closed subgroup of the <a href="Unitary_group" title="Unitary group">unitary group</a> U(V) where V is a finite-dimensional complex inner product space. Its Lie algebra consists of all skew-adjoint operators X such that exp tX lies in G for all real t. Set W = V ⊕ C with the trivial action of G on the second summand. The group G acts on W⊗N , with an element u acting as u⊗N. The <a href="Commutant" class="mw-redirect" title="Commutant">commutant</a> (or centralizer algebra) is denoted by AN = EndG W⊗N. It is generated as a *-algebra by its unitary operators and its commutant is the *-algebra spanned by the operators u⊗N. The complexification GC of G consists of all operators g in GL(V) such that g⊗N commutes with AN and g acts trivially on the second summand in C. By definition it is a closed subgroup of GL(V). The defining relations (as a commutant) show that G is an algebraic subgroup. Its intersection with U(V) coincides with G, since it is a priori a larger compact group for which the irreducible representations stay irreducible and inequivalent when restricted to G. Since AN is generated by unitaries, an invertible operator g lies in GC if the unitary operator u and positive operator p in its <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> g = u ⋅ p both lie in GC. Thus u lies in G and the operator p can be written uniquely as p = exp T with T a self-adjoint operator. By the <a href="Functional_calculus" title="Functional calculus">functional calculus</a> for polynomial functions it follows that h⊗N lies in the commutant of AN if h = exp z T with z in C. In particular taking z purely imaginary, T must have the form iX with X in the Lie algebra of G. Since every finite-dimensional representation of G occurs as a direct summand of W⊗N, it is left invariant by GC and thus every finite-dimensional representation of G extends uniquely to GC. The extension is compatible with the polar decomposition. Finally the polar decomposition implies that G is a maximal compact subgroup of GC, since a strictly larger compact subgroup would contain all integer powers of a positive operator p, a closed infinite discrete subgroup.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Decompositions_in_the_Chevalley_complexification">Decompositions in the Chevalley complexification</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Cartan_decomposition">Cartan decomposition</h3></div>

The decomposition derived from the polar decomposition

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G_{\mathbf {C} }=G\cdot P=G\cdot \exp i{\mathfrak {g}},}}</annotation>

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</math><img src="./8b28e686c448c4a96d6f711c388a70ffe2ba9a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.939ex; height:2.509ex;" alt="{\displaystyle \displaystyle {G_{\mathbf {C} }=G\cdot P=G\cdot \exp i{\mathfrak {g}},}}" loading="lazy"></dd></dl>

where 𝖌 is the Lie algebra of G, is called the <a href="Cartan_decomposition" title="Cartan decomposition">Cartan decomposition</a> of GC. The exponential factor P is invariant under conjugation by G but is not a subgroup. The complexification is invariant under taking adjoints, since G consists of unitary operators and P of positive operators.

<div class="mw-heading mw-heading3"><h3 id="Gauss_decomposition">Gauss decomposition</h3></div>

The Gauss decomposition is a generalization of the <a href="LU_decomposition" title="LU decomposition">LU decomposition</a> for the general linear group and a specialization of the <a href="Bruhat_decomposition" title="Bruhat decomposition">Bruhat decomposition</a>. For GL(V) it states that with respect to a given orthonormal basis e1, ..., en an element g of GL(V) can be factorized in the form

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {g=XDY}}</annotation>

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</math><img src="./5f04fdd165d6f5a4af2799e0f7e1c41f0db73b2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.892ex; height:2.509ex;" alt="{\displaystyle \displaystyle {g=XDY}}" loading="lazy"></dd></dl>

with X lower <a href="Unitriangular_matrix" class="mw-redirect" title="Unitriangular matrix">unitriangular</a>, Y upper unitriangular and D diagonal if and only if all the <a href="Principal_minor" class="mw-redirect" title="Principal minor">principal minors</a> of g are non-vanishing. In this case X, Y and D are uniquely determined.

In fact Gaussian elimination shows there is a unique X such that X−1 g is upper triangular.<a href="#cite_note-6">[6]</a>

The upper and lower unitriangular matrices, N+ and N−, are closed unipotent subgroups of GL(V). Their Lie algebras consist of upper and lower strictly triangular matrices. The exponential mapping is a polynomial mapping from the Lie algebra to the corresponding subgroup by nilpotence. The inverse is given by the logarithm mapping which by unipotence is also a polynomial mapping. In particular there is a correspondence between closed connected subgroups of N± and subalgebras of their Lie algebras. The exponential map is onto in each case, since the polynomial function log ( eA eB ) lies in a given Lie subalgebra if A and B do and are sufficiently small.<a href="#cite_note-7">[7]</a>

The Gauss decomposition can be extended to complexifications of other closed connected subgroups G of U(V) by using the root decomposition to write the complexified Lie algebra as<a href="#cite_note-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {{\mathfrak {g}}_{\mathbf {C} }={\mathfrak {n}}_{-}\oplus {\mathfrak {t}}_{\mathbf {C} }\oplus {\mathfrak {n}}_{+},}}</annotation>

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</math><img src="./fccc6b2c3e875520973999b0f6d8f3611f101d61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.04ex; height:2.509ex;" alt="{\displaystyle \displaystyle {{\mathfrak {g}}_{\mathbf {C} }={\mathfrak {n}}_{-}\oplus {\mathfrak {t}}_{\mathbf {C} }\oplus {\mathfrak {n}}_{+},}}" loading="lazy"></dd></dl>

where 𝖙 is the Lie algebra of a maximal torus T of G and 𝖓± are the direct sum of the corresponding positive and negative root spaces. In the weight space decomposition of V as eigenspaces of T, 𝖙 acts as diagonally, 𝖓+ acts as lowering operators and 𝖓− as raising operators. 𝖓± are nilpotent Lie algebras acting as nilpotent operators; they are each other's adjoints on V. In particular T acts by conjugation of 𝖓+, so that 𝖙C ⊕ 𝖓+ is a semidirect product of a nilpotent Lie algebra by an abelian Lie algebra.

By <a href="Engel's_theorem" title="Engel's theorem">Engel's theorem</a>, if 𝖆 ⊕ 𝖓 is a semidirect product, with 𝖆 abelian and 𝖓 nilpotent, acting on a finite-dimensional vector space W with operators in 𝖆 diagonalizable and operators in 𝖓 nilpotent, there is a vector w that is an eigenvector for 𝖆 and is annihilated by 𝖓. In fact it is enough to show there is a vector annihilated by 𝖓, which follows by induction on dim 𝖓, since the derived algebra 𝖓' annihilates a non-zero subspace of vectors on which 𝖓 / 𝖓' and 𝖆 act with the same hypotheses.

Applying this argument repeatedly to 𝖙C ⊕ 𝖓+ shows that there is an orthonormal basis e1, ..., en of V consisting of eigenvectors of 𝖙C with 𝖓+ acting as upper triangular matrices with zeros on the diagonal.

If N± and TC are the complex Lie groups corresponding to 𝖓+ and 𝖙C, then the Gauss decomposition states that the subset

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {N_{-}T_{\mathbf {C} }N_{+}}}</annotation>

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</math><img src="./515c994afaa8eb4cf1a4675c67561c5fdf2d26c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.709ex; height:2.509ex;" alt="{\displaystyle \displaystyle {N_{-}T_{\mathbf {C} }N_{+}}}" loading="lazy"></dd></dl>

is a direct product and consists of the elements in GC for which the principal minors are non-vanishing. It is open and dense. Moreover, if T denotes the maximal torus in U(V),

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {N_{\pm }=\mathbf {N} _{\pm }\cap G_{\mathbf {C} },\,\,\,T_{\mathbf {C} }=\mathbf {T} _{\mathbf {C} }\cap G_{\mathbf {C} }.}}</annotation>

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</math><img src="./4c35867338ceed43b3abf672acc4c5b51551786f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.445ex; height:2.509ex;" alt="{\displaystyle \displaystyle {N_{\pm }=\mathbf {N} _{\pm }\cap G_{\mathbf {C} },\,\,\,T_{\mathbf {C} }=\mathbf {T} _{\mathbf {C} }\cap G_{\mathbf {C} }.}}" loading="lazy"></dd></dl>

These results are an immediate consequence of the corresponding results for GL(V).<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Bruhat_decomposition">Bruhat decomposition</h3></div>

If W = NG(T) / T denotes the <a href="Weyl_group" title="Weyl group">Weyl group</a> of T and B denotes the <a href="Borel_subgroup" title="Borel subgroup">Borel subgroup</a> TC N+, the Gauss decomposition is also a consequence of the more precise <a href="Bruhat_decomposition" title="Bruhat decomposition">Bruhat decomposition</a>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G_{\mathbf {C} }=\bigcup _{\sigma \in W}B\sigma B,}}</annotation>

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</math><img src="./541d0ab51a35a429e8fa91435610fdd7c00f4b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.173ex; height:5.676ex;" alt="{\displaystyle \displaystyle {G_{\mathbf {C} }=\bigcup _{\sigma \in W}B\sigma B,}}" loading="lazy"></dd></dl>

decomposing GC into a disjoint union of <a href="Double_coset" title="Double coset">double cosets</a> of B. The complex dimension of a double coset BσB is determined by the length of σ as an element of W. The dimension is maximized at the <a href="Coxeter_element" title="Coxeter element">Coxeter element</a> and gives the unique open dense double coset. Its inverse conjugates B into the Borel subgroup of lower triangular matrices in GC.<a href="#cite_note-10">[10]</a>

The Bruhat decomposition is easy to prove for SL(n,C).<a href="#cite_note-11">[11]</a> Let B be the Borel subgroup of upper triangular matrices and TC the subgroup of diagonal matrices. So N(TC) / TC = Sn. For g in SL(n,C), take b in B so that bg maximizes the number of zeros appearing at the beginning of its rows. Because a multiple of one row can be added to another, each row has a different number of zeros in it. Multiplying by a matrix w in N(TC), it follows that wbg lies in B. For uniqueness, if w1b w2 = b0, then the entries of w1w2 vanish below the diagonal. So the product lies in TC, proving uniqueness.

<a href="#CITEREFChevalley1955">Chevalley (1955)</a> showed that the expression of an element g as g = b1σb2 becomes unique if b1 is restricted to lie in the upper unitriangular subgroup Nσ = N+ ∩ σ N− σ−1. In fact, if Mσ = N+ ∩ σ N+ σ−1, this follows from the identity

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {N_{+}=N_{\sigma }\cdot M_{\sigma }.}}</annotation>

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</math><img src="./2e9a1001c15cedbb9ca5f1d8c16bf3a9611918aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.267ex; height:2.509ex;" alt="{\displaystyle \displaystyle {N_{+}=N_{\sigma }\cdot M_{\sigma }.}}" loading="lazy"></dd></dl>

The group N+ has a natural filtration by normal subgroups N+(k) with zeros in the first k − 1 superdiagonals and the successive quotients are Abelian. Defining Nσ(k) and Mσ(k) to be the intersections with N+(k), it follows by decreasing induction on k that N+(k) = Nσ(k) ⋅ Mσ(k). Indeed, Nσ(k)N+(k + 1) and Mσ(k)N+(k + 1) are specified in N+(k) by the vanishing of complementary entries (i, j) on the kth superdiagonal according to whether σ preserves the order i < j or not.<a href="#cite_note-12">[12]</a>

The Bruhat decomposition for the other classical simple groups can be deduced from the above decomposition using the fact that they are fixed point subgroups of folding automorphisms of SL(n,C).<a href="#cite_note-13">[13]</a> For Sp(n,C), let J be the n × n matrix with 1's on the antidiagonal and 0's elsewhere and set

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {A={\begin{pmatrix}0&J\\-J&0\end{pmatrix}}.}}</annotation>

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</math><img src="./40f6bdfc247a9c4f7e093b3786b132d7462b046a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.734ex; height:6.176ex;" alt="{\displaystyle \displaystyle {A={\begin{pmatrix}0&J\\-J&0\end{pmatrix}}.}}" loading="lazy"></dd></dl>

Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (gt)−1 A−1 of SL(2n,C). It leaves the subgroups N±, TC and B invariant. If the basis elements are indexed by n, n−1, ..., 1, −1, ..., −n, then the Weyl group of Sp(n,C) consists of σ satisfying

σ(j) = −j, i.e. commuting with θ. Analogues of B, TC and N± are defined by intersection with Sp(n,C), i.e. as fixed points of θ. The uniqueness of the decomposition g = nσb = θ(n) θ(σ) θ(b) implies the Bruhat decomposition for Sp(n,C).

The same argument works for SO(n,C). It can be realised as the fixed points of ψ(g) = B (gt)−1 B−1 in SL(n,C) where B = J.

<div class="mw-heading mw-heading3"><h3 id="Iwasawa_decomposition">Iwasawa decomposition</h3></div>

The <a href="Iwasawa_decomposition" title="Iwasawa decomposition">Iwasawa decomposition</a>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G_{\mathbf {C} }=G\cdot A\cdot N}}</annotation>

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</math><img src="./eec726b8202a99db8e99f4a6ad7212ad56ca6b27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.515ex; height:2.509ex;" alt="{\displaystyle \displaystyle {G_{\mathbf {C} }=G\cdot A\cdot N}}" loading="lazy"></dd></dl>

gives a decomposition for GC for which, unlike the Cartan decomposition, the direct factor A ⋅ N is a closed subgroup, but it is no longer invariant under conjugation by G. It is the <a href="Semidirect_product" title="Semidirect product">semidirect product</a> of the <a href="Nilpotent_group" title="Nilpotent group">nilpotent</a> subgroup N by the Abelian subgroup A.

For U(V) and its complexification GL(V), this decomposition can be derived as a restatement of the <a href="Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt orthonormalization process</a>.<a href="#cite_note-14">[14]</a>

In fact let e1, ..., en be an orthonormal basis of V and let g be an element in GL(V). Applying the Gram–Schmidt process to ge1, ..., gen, there is a unique orthonormal basis f1, ..., fn and positive constants ai such that

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {f_{i}=a_{i}ge_{i}+\sum _{j<i}n_{ji}ge_{j}.}}</annotation>

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If k is the unitary taking (ei) to (fi), it follows that g−1k lies in the subgroup AN, where A is the subgroup of positive diagonal matrices with respect to (ei) and N is the subgroup of upper <a href="Unitriangular_matrix" class="mw-redirect" title="Unitriangular matrix">unitriangular matrices</a>.<a href="#cite_note-15">[15]</a>

Using the notation for the Gauss decomposition, the subgroups in the Iwasawa decomposition for GC are defined by

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {A=\exp i{\mathfrak {t}}=\mathbf {A} \cap G_{\mathbf {C} },\,\,\,N=\exp {\mathfrak {n}}_{+}=\mathbf {N} \cap G_{\mathbf {C} }.}}</annotation>

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</math><img src="./7ae11a140dc560528dc4cab9a99aa8840540aa30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.395ex; height:2.509ex;" alt="{\displaystyle \displaystyle {A=\exp i{\mathfrak {t}}=\mathbf {A} \cap G_{\mathbf {C} },\,\,\,N=\exp {\mathfrak {n}}_{+}=\mathbf {N} \cap G_{\mathbf {C} }.}}" loading="lazy"></dd></dl>

Since the decomposition is direct for GL(V), it is enough to check that GC = GAN. From the properties of the Iwasawa decomposition for GL(V), the map G × A × N is a diffeomorphism onto its image in GC, which is closed. On the other hand, the dimension of the image is the same as the dimension of GC, so it is also open. So GC = GAN because GC is connected.<a href="#cite_note-17">[17]</a>

<a href="#CITEREFZhelobenko1973">Zhelobenko (1973)</a> gives a method for explicitly computing the elements in the decomposition.<a href="#cite_note-18">[18]</a> For g in GC set h = g*g. This is a positive self-adjoint operator so its principal minors do not vanish. By the Gauss decomposition, it can therefore be written uniquely in the form

h = XDY with X in N−, D in TC and Y in N+. Since h is self-adjoint, uniqueness forces Y = X*. Since it is also positive D must lie in A and have the form D = exp iT for some unique T in 𝖙. Let a = exp iT/2 be its unique square root in A. Set n = Y and k = g n−1 a−1. Then k is unitary, so is in G, and g = kan.

<div class="mw-heading mw-heading2"><h2 id="Complex_structures_on_homogeneous_spaces">Complex structures on homogeneous spaces</h2></div>

The Iwasawa decomposition can be used to describe complex structures on the G-<a href="Orbit" title="Orbit">orbit</a>s in <a href="Complex_projective_space" title="Complex projective space">complex projective space</a> of <a href="Highest_weight_vector" class="mw-redirect" title="Highest weight vector">highest weight vectors</a> of finite-dimensional <a href="Irreducible_representation" title="Irreducible representation">irreducible representations</a> of G. In particular the identification between G / T and GC / B can be used to formulate the <a href="Borel%E2%80%93Weil_theorem" class="mw-redirect" title="Borel–Weil theorem">Borel–Weil theorem</a>. It states that each irreducible representation

of G can be obtained by <a href="Induced_representation" title="Induced representation">holomorphic induction</a> from a character of T, or equivalently that it is realized in the space of <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> of a <a href="Holomorphic_line_bundle" class="mw-redirect" title="Holomorphic line bundle">holomorphic line bundle</a> on G / T.

The closed connected subgroups of G containing T are described by <a href="Borel%E2%80%93de_Siebenthal_theory" title="Borel–de Siebenthal theory">Borel–de Siebenthal theory</a>. They are exactly the <a href="Centralizer" class="mw-redirect" title="Centralizer">centralizers</a> of tori S ⊆ T. Since every torus is generated topologically by a single element x, these are the same as centralizers CG(X) of elements X in 𝖙. By a result of Hopf CG(x) is always connected: indeed any element y is along with S contained in some maximal torus, necessarily contained in CG(x).

Given an irreducible finite-dimensional representation Vλ with highest weight vector v of weight λ, the stabilizer of C v in G is a closed subgroup H. Since v is an eigenvector of T, H contains T. The complexification GC also acts on V and the stabilizer is a closed complex subgroup P containing TC. Since v is annihilated by every raising operator corresponding to a positive root α, P contains the Borel subgroup B. The vector v is also a highest weight vector for the copy of sl2 corresponding to α, so it is annihilated by the lowering operator generating 𝖌−α if (λ, α) = 0. The Lie algebra p of P is the direct sum of 𝖙C and root space vectors annihilating v, so that

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {{\mathfrak {p}}={\mathfrak {b}}\oplus \bigoplus _{(\alpha ,\lambda )=0}{\mathfrak {g}}_{-\alpha }.}}</annotation>

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The Lie algebra of H = P ∩ G is given by p ∩ 𝖌. By the Iwasawa decomposition GC = GAN. Since AN fixes C v, the G-orbit of v in the complex projective space of Vλ coincides with the GC orbit and

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G/H=G_{\mathbf {C} }/P.}}</annotation>

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In particular

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G/T=G_{\mathbf {C} }/B.}}</annotation>

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</math><img src="./cfa697662399428d27fd958157e05066bbd9e64e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.722ex; height:2.843ex;" alt="{\displaystyle \displaystyle {G/T=G_{\mathbf {C} }/B.}}" loading="lazy"></dd></dl>

Using the identification of the Lie algebra of T with its dual, H equals the centralizer of λ in G, and hence is connected. The group P is also connected. In fact the space G / H is simply connected,

since it can be written as the quotient of the (compact) universal covering group of the compact semisimple group G / Z by a connected subgroup, where Z is the center of G.<a href="#cite_note-19">[19]</a> If Po is the identity component of P, GC / P has GC / Po as a covering space, so that P = Po. The homogeneous space GC / P has a complex structure, because P is a complex subgroup. The orbit in complex projective space is closed in the Zariski topology by <a href="Algebraic_geometry_and_analytic_geometry#Chow.27s_theorem" title="Algebraic geometry and analytic geometry">Chow's theorem</a>, so is a smooth projective variety. The Borel–Weil theorem and its generalizations are discussed in this context in <a href="#CITEREFSerre1954">Serre (1954)</a>, <a href="#CITEREFHelgason1994">Helgason (1994)</a>, <a href="#CITEREFDuistermaatKolk2000">Duistermaat & Kolk (2000)</a> and <a href="#CITEREFSepanski2007">Sepanski (2007)</a>.

The parabolic subgroup P can also be written as a union of double cosets of B

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {P=\bigcup _{\sigma \in W_{\lambda }}B\sigma B,}}</annotation>

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</math><img src="./4cf77d9ceed9edcd4e74bafec94a60a6b84fc0ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.265ex; height:6.009ex;" alt="{\displaystyle \displaystyle {P=\bigcup _{\sigma \in W_{\lambda }}B\sigma B,}}" loading="lazy"></dd></dl>

where Wλ is the stabilizer of λ in the Weyl group W. It is generated by the reflections corresponding to the simple roots orthogonal to λ.<a href="#cite_note-20">[20]</a>

<div class="mw-heading mw-heading2"><h2 id="Noncompact_real_forms">Noncompact real forms</h2></div>

There are other closed subgroups of the complexification of a compact connected Lie group G which have the same complexified Lie algebra. These are the other real forms of GC.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading3"><h3 id="Involutions_of_simply_connected_compact_Lie_groups">Involutions of simply connected compact Lie groups</h3></div>

If G is a simply connected compact Lie group and σ is an automorphism of order 2, then the fixed point subgroup K = Gσ is automatically connected. (In fact this is true for any automorphism of G, as shown for inner automorphisms by <a href="Robert_Steinberg" title="Robert Steinberg">Steinberg</a> and in general by <a href="Armand_Borel" title="Armand Borel">Borel</a>.) <a href="#cite_note-22">[22]</a>

This can be seen most directly when the involution σ corresponds to a <a href="Hermitian_symmetric_space" title="Hermitian symmetric space">Hermitian symmetric space</a>. In that case σ is inner and implemented by an element in a one-parameter subgroup exp tT contained in the center of Gσ. The innerness of σ implies that K contains a maximal torus of G, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torus S of elements exp tT is connected, since if x is any element in K there is a maximal torus containing x and S, which lies in the centralizer. On the other hand, it contains K since S is central in K and is contained in K since z lies in S. So K is the centralizer of S and hence connected. In particular K contains the center of G.<a href="#cite_note-23">[23]</a>

For a general involution σ, the connectedness of Gσ can be seen as follows.<a href="#cite_note-24">[24]</a>

The starting point is the Abelian version of the result: if T is a maximal torus of a simply connected group G and σ is an involution leaving invariant T and a choice of positive roots (or equivalently a <a href="Weyl_chamber" class="mw-redirect" title="Weyl chamber">Weyl chamber</a>), then the fixed point subgroup Tσ is connected. In fact the kernel of the exponential map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

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</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy"> onto T is a lattice Λ with a Z-basis indexed by simple roots, which σ permutes. Splitting up according to orbits, T can be written as a product of terms T on which σ acts trivially or terms T2 where σ interchanges the factors. The fixed point subgroup just corresponds to taking the diagonal subgroups in the second case, so is connected.

Now let x be any element fixed by σ, let S be a maximal torus in CG(x)σ and let T be the identity component of CG(x, S). Then T is a maximal torus in G containing x and S. It is invariant under σ and the identity component of Tσ is S. In fact since x and S commute, they are contained in a maximal torus which, because it is connected, must lie in T. By construction T is invariant under σ. The identity component of Tσ contains S, lies in CG(x)σ and centralizes S, so it equals S. But S is central in T, to T must be Abelian and hence a maximal torus. For σ acts as multiplication by −1 on the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}\ominus {\mathfrak {s}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}\ominus {\mathfrak {s}}}</annotation>

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</math><img src="./05dc0105a9db5b94e377f6d9218cf5f7380294a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.68ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {t}}\ominus {\mathfrak {s}}}" loading="lazy">, so it and therefore also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

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</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy"> are Abelian.

The proof is completed by showing that σ preserves a Weyl chamber associated with T. For then Tσ is connected so must equal S. Hence x lies in S. Since x was arbitrary, Gσ must therefore be connected.

To produce a Weyl chamber invariant under σ, note that there is no root space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}_{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}_{\alpha }}</annotation>

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</math><img src="./19ca5ac242eb190abe22a20ff71139b10d7b935a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {g}}_{\alpha }}" loading="lazy"> on which both x and S acted trivially, for this would contradict the fact that CG(x, S) has the same Lie algebra as T. Hence there must be an element s in S such that t = xs acts non-trivially on each root space. In this case t is a regular element of T—the identity component of its centralizer in G equals T. There is a unique <a href="Affine_Weyl_group" class="mw-redirect" title="Affine Weyl group">Weyl alcove</a> A in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

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</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy"> such that t lies in exp A and 0 lies in the closure of A. Since t is fixed by σ, the alcove is left invariant by σ and hence so also is the <a href="Weyl_chamber" class="mw-redirect" title="Weyl chamber">Weyl chamber</a> C containing it.

<div class="mw-heading mw-heading3"><h3 id="Conjugations_on_the_complexification">Conjugations on the complexification</h3></div>

Let G be a simply connected compact Lie group with complexification GC. The map c(g) = (g*)−1 defines an automorphism of GC as a real Lie group with G as fixed point subgroup. It is conjugate-linear on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}</annotation>

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</math><img src="./e8ddd1df6737ab81249938e37acb46f5e07ee64f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.77ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}" loading="lazy"> and satisfies c2 = id. Such automorphisms of either GC or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}</annotation>

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</math><img src="./e8ddd1df6737ab81249938e37acb46f5e07ee64f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.77ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}" loading="lazy"> are called conjugations.

Since GC is also simply connected any conjugation c1 on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}</annotation>

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</math><img src="./e8ddd1df6737ab81249938e37acb46f5e07ee64f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.77ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}" loading="lazy"> corresponds to a unique automorphism c1 of GC.

The classification of conjugations c0 reduces to that of involutions σ of G because

given a c1 there is an automorphism φ of the complex group GC such that

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {c_{0}=\varphi \circ c_{1}\circ \varphi ^{-1}}}</annotation>

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</math><img src="./87600359b9bad9a8ce2347b948ba7fe722718558.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.983ex; height:3.176ex;" alt="{\displaystyle \displaystyle {c_{0}=\varphi \circ c_{1}\circ \varphi ^{-1}}}" loading="lazy"></dd></dl>

commutes with c. The conjugation c0 then leaves G invariant and restricts to an involutive automorphism σ. By simple connectivity the same is true at the level of Lie algebras. At the Lie algebra level c0 can be recovered from σ by the formula

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {c_{0}(X+iY)=\sigma (X)-i\sigma (Y)}}</annotation>

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</math><img src="./3cec021fd19ef8975a4326237506e1e250513cb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.039ex; height:2.843ex;" alt="{\displaystyle \displaystyle {c_{0}(X+iY)=\sigma (X)-i\sigma (Y)}}" loading="lazy"></dd></dl>

for X, Y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">.

To prove the existence of φ let ψ = c1c an automorphism of the complex group GC. On the Lie algebra level it defines a self-adjoint operator for the complex inner product

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {(X,Y)=-B(X,c(Y)),}}</annotation>

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</math><img src="./94b0ce090913011b86fb116774c8ff423fafcf12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.327ex; height:2.843ex;" alt="{\displaystyle \displaystyle {(X,Y)=-B(X,c(Y)),}}" loading="lazy"></dd></dl>

where B is the <a href="Killing_form" title="Killing form">Killing form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}</annotation>

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</math><img src="./e8ddd1df6737ab81249938e37acb46f5e07ee64f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.77ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {g}}_{\mathbf {C} }}" loading="lazy">. Thus ψ2 is a positive operator and an automorphism along with all its real powers. In particular take

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\varphi =(\psi ^{2})^{1/4}}}</annotation>

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</math><img src="./54658a9ec40e7c9da473958e23355c74f954b4bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.694ex; height:3.343ex;" alt="{\displaystyle \displaystyle {\varphi =(\psi ^{2})^{1/4}}}" loading="lazy"></dd></dl>

It satisfies

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {c_{0}c=\varphi c_{1}\varphi ^{-1}c=\varphi cc_{1}\varphi =(\psi ^{2})^{1/2}\psi ^{-1}=\varphi ^{-1}cc_{1}\varphi ^{-1}=c\varphi c_{1}\varphi ^{-1}=cc_{0}.}}</annotation>

</semantics>

</math><img src="./74bb65cf23c2edcfde32a021f081b6e70c823674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.058ex; height:3.343ex;" alt="{\displaystyle \displaystyle {c_{0}c=\varphi c_{1}\varphi ^{-1}c=\varphi cc_{1}\varphi =(\psi ^{2})^{1/2}\psi ^{-1}=\varphi ^{-1}cc_{1}\varphi ^{-1}=c\varphi c_{1}\varphi ^{-1}=cc_{0}.}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Cartan_decomposition_in_a_real_form">Cartan decomposition in a real form</h3></div>

For the complexification GC, the <a href="Cartan_decomposition" title="Cartan decomposition">Cartan decomposition</a> is described above. Derived from the <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> in the complex <a href="General_linear_group" title="General linear group">general linear group</a>, it gives a diffeomorphism

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G_{\mathbf {C} }=G\cdot \exp i{\mathfrak {g}}=G\cdot P=P\cdot G.}}</annotation>

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</math><img src="./7755e16c93eec3168ba6658e0288b8d909f862bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.289ex; height:2.509ex;" alt="{\displaystyle \displaystyle {G_{\mathbf {C} }=G\cdot \exp i{\mathfrak {g}}=G\cdot P=P\cdot G.}}" loading="lazy"></dd></dl>

On GC there is a conjugation operator c corresponding to G as well as an involution σ commuting with c. Let c0 = c σ and let G0 be the fixed point subgroup of c. It is closed in the matrix group GC and therefore a Lie group. The involution σ acts on both G and G0. For the Lie algebra of G there is a decomposition

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}}</annotation>

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</math><img src="./e2d7d28fc006e33b047f0f28c4a6a7e99aceab45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.178ex; height:2.509ex;" alt="{\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}}" loading="lazy"></dd></dl>

into the +1 and −1 eigenspaces of σ. The fixed point subgroup K of σ in G is connected since G is simply connected. Its Lie algebra is the +1 eigenspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}}</annotation>

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</math><img src="./a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {k}}}" loading="lazy">. The Lie algebra of G0 is given by

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}}</annotation>

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</math><img src="./e2d7d28fc006e33b047f0f28c4a6a7e99aceab45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.178ex; height:2.509ex;" alt="{\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}}" loading="lazy"></dd></dl>

and the fixed point subgroup of σ is again K, so that G ∩ G0 = K. In G0, there is a Cartan decomposition

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G_{0}=K\cdot \exp i{\mathfrak {p}}=K\cdot P_{0}=P_{0}\cdot K}}</annotation>

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</math><img src="./989128c0313c909a712b4d137028aeaf0e860bc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.409ex; height:2.509ex;" alt="{\displaystyle \displaystyle {G_{0}=K\cdot \exp i{\mathfrak {p}}=K\cdot P_{0}=P_{0}\cdot K}}" loading="lazy"></dd></dl>

which is again a diffeomorphism onto the direct and corresponds to the polar decomposition of matrices.

It is the restriction of the decomposition on GC. The product gives a diffeomorphism onto a closed subset of G0. To check that it is surjective, for g in G0 write g = u ⋅ p with u in G and p in P. Since c0 g = g, uniqueness implies that σu = u and σp = p−1. Hence u lies in K and p in P0.

The Cartan decomposition in G0 shows that G0 is connected, simply connected and noncompact, because of the direct factor P0. Thus G0 is a noncompact real semisimple Lie group.<a href="#cite_note-25">[25]</a>

Moreover, given a maximal Abelian subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">, A = exp <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> is a toral subgroup such that σ(a) = a−1 on A; and any two such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">'s are conjugate by an element of K.

The properties of A can be shown directly. A is closed because the closure of A is a toral subgroup satisfying σ(a) = a−1, so its Lie algebra lies in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> and hence equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> by maximality. A can be generated topologically by a single element exp X, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> is the centralizer of X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">. In the K-orbit of any element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> there is an element Y such that (X,Ad k Y) is minimized at k = 1. Setting k = exp tT with T in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}}</annotation>

</semantics>

</math><img src="./a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {k}}}" loading="lazy">, it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> is the union of the conjugates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">. In particular some conjugate of X lies in any other choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">, which centralizes that conjugate; so by maximality the only possibilities are conjugates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">.<a href="#cite_note-26">[26]</a>

A similar statements hold for the action of K on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{0}=i{\mathfrak {a}}}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{0}=i{\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./c25908c3d3d724658f4a7a9c28cdb765d77377b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.28ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}_{0}=i{\mathfrak {a}}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{0}}</annotation>

</semantics>

</math><img src="./71c07b225760ae2da93b58581b8da3105b42b19c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{0}}" loading="lazy">. Morevoer, from the Cartan decomposition for G0, if A0 = exp <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{0}}</annotation>

</semantics>

</math><img src="./5db87213ff7a2ba6fc50dba0e6bd4f97a11e45f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{0}}" loading="lazy">, then

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle {G_{0}=KA_{0}K.}}</annotation>

</semantics>

</math><img src="./93f4cfbe27b3b92706db15d0f7a152bdf7e87eff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.555ex; height:2.509ex;" alt="{\displaystyle \displaystyle {G_{0}=KA_{0}K.}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Iwasawa_decomposition_in_a_real_form">Iwasawa decomposition in a real form</h3></div>

<ul><li><a href="Real_form_(Lie_theory)" title="Real form (Lie theory)">Real form (Lie theory)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width reflist-columns-2">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> See:

<ul><li><a href="#CITEREFHochschild1965">Hochschild 1965</a></li>

<li><a href="#CITEREFBourbaki1981">Bourbaki 1981</a>, pp. 212–214</li></ul>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFBourbaki1981">Bourbaki 1981</a>, pp. 210–214

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFHochschild1966">Hochschild 1966</a>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> See:

<ul><li><a href="#CITEREFHochschild1965">Hochschild 1965</a></li>

<li><a href="#CITEREFChevalley1946">Chevalley 1946</a></li>

<li><a href="#CITEREFBröckertom_Dieck1985">Bröcker & tom Dieck 1985</a></li></ul>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> See:

<ul><li><a href="#CITEREFChevalley1946">Chevalley 1946</a></li>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFZhelobenko1973">Zhelobenko 1973</a>, p. 28

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFBump2004">Bump 2004</a>, pp. 202–203

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> See:

<li><a href="#CITEREFZhelobenko1973">Zhelobenko 1973</a></li></ul>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFZhelobenko1973">Zhelobenko 1973</a>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> See:

<ul><li><a href="#CITEREFGelfandNaimark1950">Gelfand & Naimark 1950</a>, section 18, for SL(n,C)</li>

<li><a href="#CITEREFBruhat1956">Bruhat 1956</a>, p. 187 for SO(n,C) and Sp(n,C)</li>

<li><a href="#CITEREFChevalley1955">Chevalley 1955</a> for complexifications of simple compact Lie groups</li>

<li><a href="#CITEREFHelgason1978">Helgason 1978</a>, pp. 403–406 for <a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a>'s method</li>

<li><a href="#CITEREFHumphreys1981">Humphreys 1981</a> for a treatment using algebraic groups</li>

<li><a href="#CITEREFCarter1972">Carter 1972</a>, Chapter 8</li>

<li><a href="#CITEREFDieudonné1977">Dieudonné 1977</a>, pp. 216–217</li>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFSteinberg1974">Steinberg 1974</a>, p. 73

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFChevalley1955">Chevalley 1955</a>, p. 41

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> See:

<ul><li><a href="#CITEREFSteinberg1974">Steinberg 1974</a>, pp. 73–74</li>

<li><a href="#CITEREFBourbaki1981a">Bourbaki 1981a</a>, pp. 53–54</li></ul>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFSepanski2007">Sepanski 2007</a>, p. 8

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFKnapp2001">Knapp 2001</a>, p. 117

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> See:

<ul><li><a href="#CITEREFZhelobenko1973">Zhelobenko 1973</a>, pp. 288–290</li>

<li><a href="#CITEREFDieudonné1977">Dieudonné 1977</a>, pp. 197–207</li>

<li><a href="#CITEREFHelgason1978">Helgason 1978</a>, pp. 257–262</li>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFBump2004">Bump 2004</a>, pp. 203–204

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFZhelobenko1973">Zhelobenko 1973</a>, p. 289

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFHelgason1978">Helgason 1978</a>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> See:

<ul><li><a href="#CITEREFHumphreys1981">Humphreys 1981</a></li>

<li><a href="#CITEREFBourbaki1981a">Bourbaki 1981a</a></li></ul>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFDieudonné1977">Dieudonné 1977</a>, pp. 164–173

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> See:

<ul><li><a href="#CITEREFHelgason1978">Helgason 1978</a>, pp. 320–321</li>

<li><a href="#CITEREFBourbaki1982">Bourbaki 1982</a>, pp. 46–48</li>

<li><a href="#CITEREFDuistermaatKolk2000">Duistermaat & Kolk 2000</a>, pp. 194–195</li>

<li><a href="#CITEREFDieudonné1977">Dieudonné 1977</a>, p. 151, Exercise 11</li></ul>

</li>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> See:

<a href="#CITEREFBourbaki1982">Bourbaki 1982</a>, pp. 46–48

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFDieudonné1977">Dieudonné 1977</a>, pp. 166–168

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFHelgason1978">Helgason 1978</a>, p. 248

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBourbaki1981" class="citation cs2">Bourbaki, N. (1981), Groupes et Algèbres de Lie (Chapitre 3), Éléments de Mathématique, Hermann, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3540339403</bdi></cite></li>

<li><cite id="CITEREFBourbaki1981a" class="citation cs2">Bourbaki, N. (1981a), Groupes et Algèbres de Lie (Chapitres 4,5 et 6), Éléments de Mathématique, Masson, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-2225760761</bdi></cite></li>

<li><cite id="CITEREFBourbaki1982" class="citation cs2">Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3540343929</bdi></cite></li>

<li><cite id="CITEREFBröckertom_Dieck1985" class="citation cs2">Bröcker, T.; tom Dieck, T. (1985), Representations of Compact Lie Groups, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 98, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3540136781</bdi></cite></li>

<li><cite id="CITEREFBruhat1956" class="citation cs2">Bruhat, F. (1956), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=BSMF_1956__84__97_0">"Sur les représentations induites des groupes de Lie"</a>, Bull. Soc. Math. France, 84: 97–205, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.1469">10.24033/bsmf.1469</a></cite></li>

<li><cite id="CITEREFBump2004" class="citation cs2">Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics, vol. 225, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0387211541</bdi></cite></li>

<li><cite id="CITEREFCarter1972" class="citation cs2">Carter, Roger W. (1989) [1972], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nW9tPZUMkdIC&pg=PR1">Simple groups of Lie type</a>, Wiley Classics Library, vol. 22, Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780471506836</bdi></cite></li>

<li><cite id="CITEREFChevalley1946" class="citation cs2">Chevalley, C. (2018) [1946], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NwNKDwAAQBAJ&pg=PP1">Theory of Lie Groups I</a>, Dover, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486824536</bdi></cite></li>

<li><cite id="CITEREFChevalley1955" class="citation cs2">Chevalley, C. (1955), <a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.tmj/1178245104">"Sur certains groupes simples"</a>, <a href="T%C3%B4hoku_Mathematical_Journal" class="mw-redirect" title="Tôhoku Mathematical Journal">Tôhoku Mathematical Journal</a>, 7 (1–2): 14–66, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2748%2Ftmj%2F1178245104">10.2748/tmj/1178245104</a></cite></li>

<li><cite id="CITEREFDieudonné1977" class="citation cs2">Dieudonné, J. (1977), Compact Lie groups and semisimple Lie groups, Chapter XXI, Treatise on analysis, vol. 5, Academic Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0122155055</bdi></cite></li>

<li><cite id="CITEREFDuistermaatKolk2000" class="citation cs2">Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3540152934</bdi></cite></li>

<li><cite id="CITEREFGelfandNaimark1950" class="citation cs2 cs1-prop-foreign-lang-source">Gelfand, I. M.; Naimark, M. A. (1950), <a rel="nofollow" class="external text" href="https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=1100&option_lang=eng">"Unitary representations of the classical groups"</a>, Trudy Mat. Inst. Steklov. (in Russian), 36: 3–288</cite></li>

<li><cite id="CITEREFHelgason1978" class="citation cs2">Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0821828489</bdi></cite></li>

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