Existence and uniqueness up to isomorphism of the split real form

Given: The Axiom of Choice and a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$; after the choices justified in [L0], write $\mathfrak{h}$ for a Cartan subalgebra, $\mathrm{\Phi }$ for its root system and $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ for a base with Cartan matrix $A$; when a split real form of $\mathfrak{g}$ is under discussion, write ${\mathfrak{h}}_1$ for a Cartan subalgebra of it as in [L3].

The Axiom of Choice is The Axiom of Choice; it is inherited from the Serre presentation theorem and root data of [L1] and [L2], from the conjugacy theorem of [L4] and from the Chevalley-basis statement [L6], whose statements carry the assumption.

Every finite-dimensional complex semisimple Lie algebra has a Cartan subalgebra. For its finite reduced root system, a regular vector determines a positive system and its indecomposable positive roots form a base; the simple roots form a basis of the real root span and every root has integral coefficients of one sign (Existence of Cartan subalgebras, Positive systems and simple roots, Simple roots form a signed integral basis).

The Serre presentation theorem, in both directions: $\mathfrak{g}\cong \mathfrak{g}(A)$ through the assignment $E_i\mapsto e_i$, $F_i\mapsto f_i$, $H_i\mapsto h_i$ determined by a root ${\mathfrak{s}\mathfrak{l}}_2$ triple $(e_i,f_i,h_i)$ over the base $\mathrm{\Delta }$ (The root ${\mathfrak{s}\mathfrak{l}}_2$ triple), the elements $e_i,f_i,h_i$ generating $\mathfrak{g}$ and satisfying exactly the relations of the Serre algebra $\mathfrak{g}(A)$; conversely, for any Cartan subalgebra ${\mathfrak{h}}^{\mathrm{\prime }}$ of $\mathfrak{g}$, any base ${\mathrm{\Delta }}^{\mathrm{\prime }}$ of $\mathrm{\Phi }(\mathfrak{g},{\mathfrak{h}}^{\mathrm{\prime }})$ with Cartan matrix $A$ and any root ${\mathfrak{s}\mathfrak{l}}_2$ triples $(e_i^{\mathrm{\prime }},f_i^{\mathrm{\prime }},h_i^{\mathrm{\prime }})$ over ${\mathrm{\Delta }}^{\mathrm{\prime }}$, the elements $e_i^{\mathrm{\prime }},f_i^{\mathrm{\prime }},h_i^{\mathrm{\prime }}$ generate $\mathfrak{g}$ and satisfy exactly the relations of $\mathfrak{g}(A)$, so that $E_i\mapsto e_i^{\mathrm{\prime }}$, $F_i\mapsto f_i^{\mathrm{\prime }}$, $H_i\mapsto h_i^{\mathrm{\prime }}$ extends to an isomorphism $\mathfrak{g}(A)\to \mathfrak{g}$; a homomorphism out of $\mathfrak{g}(A)$ is defined by prescribing the images of the generators and checking the relations (Serre presentation theorem, Serre Lie algebra of a finite-type Cartan matrix, Lie algebra presented by generators and relations, The root ${\mathfrak{s}\mathfrak{l}}_2$ triple).

Let ${\mathfrak{h}}^{\mathrm{\prime }}$ be a Cartan subalgebra of $\mathfrak{g}$, ${\mathrm{\Phi }}^{\mathrm{\prime }}=\mathrm{\Phi }(\mathfrak{g},{\mathfrak{h}}^{\mathrm{\prime }})$ its root system and $h_{\beta }\in {\mathfrak{h}}^{\mathrm{\prime }}$ the coroot of a root $\beta$, characterized by $\beta (h_{\beta })=2$ and $B(h_{\beta },H)=2\beta (H)\mathrm{/}\beta (H_{\beta })$ for $H\in {\mathfrak{h}}^{\mathrm{\prime }}$. Then ${\mathrm{\Phi }}^{\mathrm{\prime }}$ is finite, $\mathfrak{g}={\mathfrak{h}}^{\mathrm{\prime }}\oplus {⨁}_{\beta \in {\mathrm{\Phi }}^{\mathrm{\prime }}}{\mathfrak{g}}_{\beta }$ with one-dimensional root spaces, and for a base ${\mathrm{\Delta }}^{\mathrm{\prime }}=\{{\alpha }_1^{\mathrm{\prime }},\dots ,{\alpha }_r^{\mathrm{\prime }}\}$ of ${\mathrm{\Phi }}^{\mathrm{\prime }}$ the Cartan matrix is $a_{ij}^{\mathrm{\prime }}={\alpha }_j^{\mathrm{\prime }}(h_{{\alpha }_i^{\mathrm{\prime }}})=⟨{\alpha }_j^{\mathrm{\prime }},{\alpha }_i^{\mathrm{\prime }\vee }⟩=2({\alpha }_j^{\mathrm{\prime }},{\alpha }_i^{\mathrm{\prime }})\mathrm{/}({\alpha }_i^{\mathrm{\prime }},{\alpha }_i^{\mathrm{\prime }})$, an integer determined by the inner product $(\lambda ,\mu )=B(H_{\lambda },H_{\mu })$ on ${\mathrm{span}}_{\mathbb{R}}{\mathrm{\Phi }}^{\mathrm{\prime }}$ (Roots of a complex semisimple Lie algebra form a reduced crystallographic root system, Root and root space, Coroot of a Lie-algebra root, Cartan matrix of a based root system, Cartan integers are integers).

A real form ${\mathfrak{g}}_1$ of $\mathfrak{g}$ (Real form of a complex Lie algebra) is split if it contains a Cartan subalgebra ${\mathfrak{h}}_1$ of ${\mathfrak{g}}_1$ (Cartan subalgebra) such that all operators ${\mathrm{ad}}_H$, $H\in {\mathfrak{h}}_1$, are simultaneously diagonalizable over $\mathbb{R}$; equivalently ${\mathfrak{g}}_1$ has a basis in which all these operators are diagonal with real eigenvalues (Split real form).

A subalgebra ${\mathfrak{h}}^{\mathrm{\prime }}$ of $\mathfrak{g}$ is a Cartan subalgebra if and only if it is nilpotent and $N_{\mathfrak{g}}({\mathfrak{h}}^{\mathrm{\prime }})={\mathfrak{h}}^{\mathrm{\prime }}$; every Cartan subalgebra of $\mathfrak{g}$ is abelian and satisfies $N_{\mathfrak{g}}({\mathfrak{h}}^{\mathrm{\prime }})=C_{\mathfrak{g}}({\mathfrak{h}}^{\mathrm{\prime }})={\mathfrak{h}}^{\mathrm{\prime }}$, and any two Cartan subalgebras of $\mathfrak{g}$ are carried to one another by an inner automorphism in the connected adjoint group of $\mathfrak{g}$ (Cartan subalgebra, Cartan subalgebras are exactly maximal toral subalgebras, Conjugacy of Cartan subalgebras, The Axiom of Choice).

$B$ is symmetric, invariant, nondegenerate, and preserved by every automorphism $\psi$ of $\mathfrak{g}$: $B(\psi x,\psi y)=B(x,y)$, because ${\mathrm{ad}}_{\psi x}=\psi \circ {\mathrm{ad}}_x\circ {\psi }^{-1}$ and $B$ is a trace form. For $x\in {\mathfrak{g}}_{\beta }$ and $y\in {\mathfrak{g}}_{-\beta }$ one has $[x,y]=B(x,y)H_{\beta }$, where $H_{\beta }$ is the Killing-dual of $\beta$, and the pairing ${\mathfrak{g}}_{\beta }\times {\mathfrak{g}}_{-\beta }\to \mathbb{C}$ given by $B$ is nondegenerate (Killing form, Trace forms are symmetric and invariant, Killing-dual vector of a root, Opposite root spaces pair nondegenerately, Brackets of root spaces).

The root-vector basis of $(\mathfrak{g},\mathfrak{h})$ can be rescaled so that $[e_{\alpha },e_{-\alpha }]=h_{\alpha }$ for every root and the structure constants $N_{\alpha \beta }$, defined by $[e_{\alpha },e_{\beta }]=N_{\alpha \beta }{e}_{\alpha +\beta }$, are integers satisfying $N_{\alpha \beta }=-N_{-\alpha ,-\beta }$; the real span ${\mathrm{span}}_{\mathbb{R}}\{h_{\alpha },e_{\alpha }:\alpha \in \mathrm{\Phi }\}$ is then a split real form of $\mathfrak{g}$ with integer structure constants in the basis $\{h_{{\alpha }_1},\dots ,h_{{\alpha }_r}\}\cup \{e_{\alpha }:\alpha \in \mathrm{\Phi }\}$ (Chevalley basis and real structure constants).

Proof technique: direct.

1.1 Choose a Cartan subalgebra $\mathfrak{h}$ and a base $\mathrm{\Delta }$ by [L0]. By [L6] the root vectors of this Cartan data can be rescaled so that $[e_{\alpha },e_{-\alpha }]=h_{\alpha }$, the structure constants $N_{\alpha \beta }$ are integers with $N_{\alpha \beta }=-N_{-\alpha ,-\beta }$, and ${\mathfrak{g}}_0:={\mathrm{span}}_{\mathbb{R}}\{h_{\alpha },e_{\alpha }:\alpha \in \mathrm{\Phi }\}$ is a split real form of $\mathfrak{g}$ with integer structure constants in the basis $\{h_{{\alpha }_1},\dots ,h_{{\alpha }_r}\}\cup \{e_{\alpha }\}$. Write $h_i:=h_{{\alpha }_i}$, $e_i:=e_{{\alpha }_i}$ and $f_i:=e_{-{\alpha }_i}$; then $(e_i,f_i,h_i)$ is a root ${\mathfrak{s}\mathfrak{l}}_2$ triple over $\mathrm{\Delta }$, and the real span $\mathfrak{a}$ of the iterated brackets of $h_i,e_i,f_i$ is a subalgebra of ${\mathfrak{g}}_0$ containing these generators. Its complexification is a complex subalgebra of $\mathfrak{g}$ containing $e_i,f_i,h_i$, which generate $\mathfrak{g}$ by [L1], so it equals $\mathfrak{g}$; hence ${\dim }_{\mathbb{R}}\mathfrak{a}={\dim }_{\mathbb{C}}\mathfrak{g}={\dim }_{\mathbb{R}}{\mathfrak{g}}_0$ and $\mathfrak{a}={\mathfrak{g}}_0$. In particular $\mathfrak{g}$ has a split real form, generated by the Serre triple $(e_i,f_i,h_i)$. [A1, L0, L1, L6, algebra]

1.2 Let ${\mathfrak{g}}_1$ be a split real form of $\mathfrak{g}$ with split Cartan subalgebra ${\mathfrak{h}}_1$ as in [L3], and put ${\mathfrak{h}}^{\mathrm{\prime }}:={\mathfrak{h}}_1^{\mathbb{C}}$. Then ${\mathfrak{h}}^{\mathrm{\prime }}$ is a Cartan subalgebra of $\mathfrak{g}$: it is nilpotent, because the lower central series of the real nilpotent Lie algebra ${\mathfrak{h}}_1$ complexifies to the lower central series of ${\mathfrak{h}}^{\mathrm{\prime }}$; and it is self-normalizing, because if $x=u+iv$ with $u,v\in {\mathfrak{g}}_1$ satisfies $[x,{\mathfrak{h}}^{\mathrm{\prime }}]\subseteq {\mathfrak{h}}^{\mathrm{\prime }}$, then $[u,{\mathfrak{h}}_1],[v,{\mathfrak{h}}_1]\subseteq {\mathfrak{h}}^{\mathrm{\prime }}\cap {\mathfrak{g}}_1={\mathfrak{h}}_1$, so $u,v\in N_{{\mathfrak{g}}_1}({\mathfrak{h}}_1)={\mathfrak{h}}_1$ and $x\in {\mathfrak{h}}^{\mathrm{\prime }}$. [L3, L4, algebra]

1.3 By [L4] there is an inner automorphism $\psi$ of $\mathfrak{g}$ with $\psi ({\mathfrak{h}}^{\mathrm{\prime }})=\mathfrak{h}$. The transpose ${\psi }_{\ast }$, defined by ${\psi }_{\ast }\beta =\beta \circ {\psi }^{-1}$ on ${\mathfrak{h}}^{\mathrm{\prime }}$, carries ${\mathrm{\Phi }}^{\mathrm{\prime }}:=\mathrm{\Phi }(\mathfrak{g},{\mathfrak{h}}^{\mathrm{\prime }})$ bijectively onto $\mathrm{\Phi }$, and since $B$ is $\psi$-invariant by [L5] it is an isometry for the inner products $(\lambda ,\mu )=B(H_{\lambda },H_{\mu })$ of [L2]; therefore it preserves Cartan integers $⟨\beta ,{\alpha }^{\vee }⟩=2(\beta ,\alpha )\mathrm{/}(\alpha ,\alpha )$. Hence ${\mathrm{\Delta }}^{\mathrm{\prime }}:={\psi }_{\ast }^{-1}(\mathrm{\Delta })$ is a base of ${\mathrm{\Phi }}^{\mathrm{\prime }}$: it is a linearly independent subset of ${\mathrm{\Phi }}^{\mathrm{\prime }}$ with $r$ elements, and every ${\beta }^{\mathrm{\prime }}\in {\mathrm{\Phi }}^{\mathrm{\prime }}$ is a nonnegative or nonpositive integral combination of ${\mathrm{\Delta }}^{\mathrm{\prime }}$, because ${\psi }_{\ast }{\beta }^{\mathrm{\prime }}$ is such a combination of the base $\mathrm{\Delta }$. Its Cartan matrix $A^{\mathrm{\prime }}$ equals $A$, since $a_{ij}^{\mathrm{\prime }}=⟨{\alpha }_j^{\mathrm{\prime }},{\alpha }_i^{\mathrm{\prime }\vee }⟩=⟨{\psi }_{\ast }{\alpha }_j^{\mathrm{\prime }},{\psi }_{\ast }{\alpha }_i^{\mathrm{\prime }\vee }⟩=⟨{\alpha }_j,{\alpha }_i^{\vee }⟩=a_{ij}\mathrm{.}$ [L2, L4, L5, algebra]

1.4 For every $\beta \in {\mathrm{\Phi }}^{\mathrm{\prime }}$ the space ${\mathfrak{g}}_1\cap {\mathfrak{g}}_{\beta }$ is a real line. Indeed, by [L3] the operators ${\mathrm{ad}}_H$ with $H\in {\mathfrak{h}}_1$ are simultaneously diagonalizable over $\mathbb{R}$ on ${\mathfrak{g}}_1$; let ${\mathfrak{g}}_1={⨁}_{\lambda }{V}_{\lambda }$ be the decomposition into common eigenspaces. Complexifying gives $\mathfrak{g}={⨁}_{\lambda }{V}_{\lambda }^{\mathbb{C}}$ with $V_{\lambda }^{\mathbb{C}}\subseteq {\mathfrak{g}}_{{\lambda }^{\mathbb{C}}}$, where ${\lambda }^{\mathbb{C}}$ is the $\mathbb{C}$-linear extension of $\lambda$; since ${\mathfrak{h}}^{\mathrm{\prime }}$ is a Cartan subalgebra with $C_{\mathfrak{g}}({\mathfrak{h}}^{\mathrm{\prime }})={\mathfrak{h}}^{\mathrm{\prime }}$ by [L4] and the weight space of weight $0$ is ${\mathfrak{h}}^{\mathrm{\prime }}$ by [L2], the zero eigenspace is $V_0={\mathfrak{h}}_1$, the intersection ${\mathfrak{h}}^{\mathrm{\prime }}\cap {\mathfrak{g}}_1$ being ${\mathfrak{h}}_1$. Every nonzero $\lambda$ with $V_{\lambda }\mathrm{\ne }0$ satisfies $V_{\lambda }^{\mathbb{C}}\subseteq {\mathfrak{g}}_{{\lambda }^{\mathbb{C}}}\mathrm{\ne }0$, so ${\lambda }^{\mathbb{C}}\in {\mathrm{\Phi }}^{\mathrm{\prime }}$, distinct nonzero $\lambda$ have distinct extensions ${\lambda }^{\mathbb{C}}$, and ${\dim }_{\mathbb{R}}{V}_{\lambda }={\dim }_{\mathbb{C}}{V}_{\lambda }^{\mathbb{C}}\le 1$. Hence ${\dim }_{\mathbb{R}}{\mathfrak{g}}_1={\dim }_{\mathbb{R}}{\mathfrak{h}}_1+{\sum }_{\lambda \mathrm{\ne }0}{\dim }_{\mathbb{R}}{V}_{\lambda }\le {\dim }_{\mathbb{R}}{\mathfrak{h}}_1+\mathrm{\#}\{\lambda \mathrm{\ne }0:V_{\lambda }\mathrm{\ne }0\}\le {\dim }_{\mathbb{R}}{\mathfrak{h}}_1+\mathrm{\mid }{\mathrm{\Phi }}^{\mathrm{\prime }}\mathrm{\mid }={\dim }_{\mathbb{C}}{\mathfrak{h}}^{\mathrm{\prime }}+{\dim }_{\mathbb{C}}{⨁}_{\beta \in {\mathrm{\Phi }}^{\mathrm{\prime }}}{\mathfrak{g}}_{\beta }={\dim }_{\mathbb{C}}\mathfrak{g}={\dim }_{\mathbb{R}}{\mathfrak{g}}_1,$ so all inequalities are equalities: each occurring nonzero $\lambda$ contributes a distinct root and ${\dim }_{\mathbb{R}}{V}_{\lambda }=1$, every root of ${\mathrm{\Phi }}^{\mathrm{\prime }}$ occurs, and ${\mathfrak{g}}_1\cap {\mathfrak{g}}_{\beta }=V_{\beta }$ is one-dimensional over $\mathbb{R}$. [L2, L3, L4, algebra]

2.1 Choose $0\mathrm{\ne }{x}_i\in {\mathfrak{g}}_1\cap {\mathfrak{g}}_{{\alpha }_i^{\mathrm{\prime }}}$ and let $y_i\in {\mathfrak{g}}_1\cap {\mathfrak{g}}_{-{\alpha }_i^{\mathrm{\prime }}}$ be the vector with $[x_i,y_i]=h_{{\alpha }_i^{\mathrm{\prime }}}$; it exists and is unique because by [L5] one has $[x_i,y]=B(x_i,y)H_{{\alpha }_i^{\mathrm{\prime }}}$ for $y\in {\mathfrak{g}}_{-{\alpha }_i^{\mathrm{\prime }}}$, where $H_{{\alpha }_i^{\mathrm{\prime }}}$ is the Killing-dual, and $y\mapsto B(x_i,y)$ is a nonzero $\mathbb{R}$-linear functional on the real line ${\mathfrak{g}}_1\cap {\mathfrak{g}}_{-{\alpha }_i^{\mathrm{\prime }}}$ by the nondegeneracy of the pairing in [L5]; moreover $H_{{\alpha }_i^{\mathrm{\prime }}}\in {\mathfrak{h}}_1$, since writing $H_{{\alpha }_i^{\mathrm{\prime }}}=u+iv$ with $u,v\in {\mathfrak{h}}_1$ and evaluating $B(H_{{\alpha }_i^{\mathrm{\prime }}},H)={\alpha }_i^{\mathrm{\prime }}(H)\in \mathbb{R}$ on $H\in {\mathfrak{h}}_1$ gives $B(v,{\mathfrak{h}}_1)=0$ and hence $v=0$, so the image of $y\mapsto [x_i,y]$ is the real line $\mathbb{R}{h}_{{\alpha }_i^{\mathrm{\prime }}}$. Put $z_i:=h_{{\alpha }_i^{\mathrm{\prime }}}$. Then $z_i\in {\mathfrak{h}}_1$, the relations $[z_i,x_i]={\alpha }_i^{\mathrm{\prime }}(z_i)x_i=2x_i$ and $[z_i,y_i]=-2y_i$ hold by [L2], and $[x_i,y_i]=z_i$ by construction; so $(x_i,y_i,z_i)$ is a root ${\mathfrak{s}\mathfrak{l}}_2$ triple over ${\mathrm{\Delta }}^{\mathrm{\prime }}$ with coroot $z_i$. Applying [L1] to the Cartan subalgebra ${\mathfrak{h}}^{\mathrm{\prime }}$ and the base ${\mathrm{\Delta }}^{\mathrm{\prime }}$ of ${\mathrm{\Phi }}^{\mathrm{\prime }}$, the elements $x_i,y_i,z_i$ generate $\mathfrak{g}$ and satisfy exactly the relations of $\mathfrak{g}(A^{\mathrm{\prime }})=\mathfrak{g}(A)$ (step 1.3). [L1, L2, L5, step 1.3, step 1.4]

3.1 Let $\mathfrak{g}(A)$ be the Serre algebra of $A$. By [L1] the assignment $E_i\mapsto e_i$, $F_i\mapsto f_i$, $H_i\mapsto h_i$ extends to an isomorphism ${\rho }_0:\mathfrak{g}(A)\to \mathfrak{g}$, and by [L1] and step 2.1 the assignment $E_i\mapsto x_i$, $F_i\mapsto y_i$, $H_i\mapsto z_i$ extends to a homomorphism ${\rho }_1:\mathfrak{g}(A)\to \mathfrak{g}$. The image of ${\rho }_1$ contains $x_i,y_i,z_i$, which generate $\mathfrak{g}$, so ${\rho }_1$ is surjective; since $\dim \mathfrak{g}(A)=\dim \mathfrak{g}$ by [L1], it is an isomorphism. Let $\mathfrak{g}(A{)}_{\mathbb{R}}$ be the real span of all iterated brackets of $E_i,F_i,H_i$ in $\mathfrak{g}(A)$; it is a real form of $\mathfrak{g}(A)$, because these iterated brackets span $\mathfrak{g}(A)$ over $\mathbb{C}$. Now ${\rho }_0(\mathfrak{g}(A{)}_{\mathbb{R}})$ is the real span of the iterated brackets of $e_i,f_i,h_i$, which is ${\mathfrak{g}}_0$ by step 1.1; and ${\rho }_1(\mathfrak{g}(A{)}_{\mathbb{R}})$ is the real span of the iterated brackets of $x_i,y_i,z_i$, a real subalgebra of ${\mathfrak{g}}_1$ whose complexification is the complex span of the same brackets, namely $\mathfrak{g}$ by step 2.1, so that it equals ${\mathfrak{g}}_1$. Hence the restrictions of ${\rho }_0$ and ${\rho }_1$ to $\mathfrak{g}(A{)}_{\mathbb{R}}$ are isomorphisms of real Lie algebras onto ${\mathfrak{g}}_0$ and ${\mathfrak{g}}_1$, and ${\mathfrak{g}}_0\cong {\mathfrak{g}}_1$ by composition. [L1, step 1.1, step 2.1, algebra]

4.1 By step 1.1 a split real form of $\mathfrak{g}$ exists, and by steps 1.2-3.1 every split real form ${\mathfrak{g}}_1$ of $\mathfrak{g}$ is isomorphic to ${\mathfrak{g}}_0$. Hence every finite-dimensional complex semisimple Lie algebra has a split real form and any two split real forms are isomorphic as real Lie algebras, that is, the split real form is unique up to isomorphism. [A1, step 1.1, step 3.1] ∎