Existence of a Cartan involution

The adjoint operators of ${\mathfrak{g}}_{\mathbb{R}}$ are the realifications of those of $\mathfrak{g}$, so the real trace of the realification of a complex-linear endomorphism is twice its complex trace and $B_{\mathbb{R}}(Z,W)=2\mathrm{Re}B(Z,W)$ for all $Z,W\in \mathfrak{g}$. If $Z$ lies in the radical of $B_{\mathbb{R}}$ then $\mathrm{Re}B(Z,W)=0$ for all $W$, and substituting $iW$ gives $\mathrm{Im}B(Z,W)=0$; hence $Z=0$ by nondegeneracy of $B$, so $B_{\mathbb{R}}$ is nondegenerate and ${\mathfrak{g}}_{\mathbb{R}}$ is semisimple, with every derivation inner and $Z({\mathfrak{g}}_{\mathbb{R}})=0$.

The restriction of $B$ to ${\mathfrak{g}}_0$ is $B_0$, because in a real basis of ${\mathfrak{g}}_0$ the matrices of ${\mathrm{ad}}_X$, $X\in {\mathfrak{g}}_0$, acting on $\mathfrak{g}$ have the real block form of the realification of ${\mathrm{ad}}_X$ acting on ${\mathfrak{g}}_0$. In particular $B$ is real-valued on ${\mathfrak{g}}_0\times {\mathfrak{g}}_0$, and $B_0$ is nondegenerate since ${\mathfrak{g}}_0$ is semisimple.

The map $\tau$ is real-linear on ${\mathfrak{g}}_{\mathbb{R}}$ with ${\tau }^2=\mathrm{i}\mathrm{d}$ and preserves brackets, and $B(\tau Z,\tau W)=\overline{B(Z,W)}$ for all $Z,W$: writing $Z=X+iY$, $W=X^{\mathrm{\prime }}+i{Y}^{\mathrm{\prime }}$ with $X,Y,X^{\mathrm{\prime }},Y^{\mathrm{\prime }}\in {\mathfrak{u}}_0$, both sides equal $B(X,X^{\mathrm{\prime }})-B(Y,Y^{\mathrm{\prime }})-i(B(X,Y^{\mathrm{\prime }})+B(Y,X^{\mathrm{\prime }}))$ by complex bilinearity and symmetry of $B$. Hence $B_{\mathbb{R}}(\tau Z,\tau W)=B_{\mathbb{R}}(Z,W)$.

$\tau$ is a Cartan involution of ${\mathfrak{g}}_{\mathbb{R}}$: for $Z=X+iY\mathrm{\ne }0$ with $X,Y\in {\mathfrak{u}}_0$ one has $(B_{\mathbb{R}}{)}_{\tau }(Z,Z)=-B_{\mathbb{R}}(Z,\tau Z)=-2\mathrm{Re}B(Z,\tau Z)=-2(B(X,X)+B(Y,Y))>0$, since $B(Z,\tau Z)=B(X,X)+B(Y,Y)$ and the Killing form of ${\mathfrak{u}}_0$ is negative definite. Write $B_{\tau }:=(B_{\mathbb{R}}{)}_{\tau }$ for this inner product.

Put $\omega :=\sigma \tau \in \mathrm{Aut}({\mathfrak{g}}_{\mathbb{R}})$, an invertible automorphism, and note $\tau \omega ={\omega }^{-1}\tau$, because both sides equal $\tau \sigma \tau$. Invariance of $B_{\mathbb{R}}$ under ${\omega }^{-1}$ and under $\tau$ gives $B_{\tau }(\omega Z,W)=-B_{\mathbb{R}}(\omega Z,\tau W)=-B_{\mathbb{R}}(Z,{\omega }^{-1}\tau W)=-B_{\mathbb{R}}(Z,\tau \omega W)=B_{\tau }(Z,\omega W)$ for all $Z,W$, so $\omega$ is self-adjoint for $B_{\tau }$; since $\omega$ is invertible, $\rho :={\omega }^2={\omega }^{\ast }\omega$ satisfies $B_{\tau }(\rho Z,Z)=B_{\tau }(\omega Z,\omega Z)>0$ for $Z\mathrm{\ne }0$, so $\rho$ is a self-adjoint positive definite automorphism of ${\mathfrak{g}}_{\mathbb{R}}$.

By [L4] choose a $B_{\tau }$-orthonormal eigenbasis of $\rho$ with eigenvalues ${\lambda }_j>0$ and let ${\rho }^r$, $r\in \mathbb{R}$, act as ${\lambda }_j^r$ on the eigenspace for ${\lambda }_j$. If $X,Y$ are eigenvectors with eigenvalues ${\lambda }_i,{\lambda }_j$, then $\rho [X,Y]=[\rho X,\rho Y]={\lambda }_i{\lambda }_j[X,Y]$, so ${\rho }^r[X,Y]=({\lambda }_i{\lambda }_j{)}^r[X,Y]=[{\rho }^{r}X,{\rho }^{r}Y]$ and, by bilinearity, ${\rho }^r\in \mathrm{Aut}({\mathfrak{g}}_{\mathbb{R}})$; also ${\rho }^r$ commutes with $\rho$ and with $\omega$.

Let $D$ act as $\log {\lambda }_j$ on the eigenspace for ${\lambda }_j$, so that $\exp (D)=\rho$ and $D$ is self-adjoint for $B_{\tau }$. For eigenvectors $X,Y$ as in step 4.1, $D[X,Y]=(\log {\lambda }_i+\log {\lambda }_j)[X,Y]=[DX,Y]+[X,DY]$, hence $D$ is a derivation of ${\mathfrak{g}}_{\mathbb{R}}$ by bilinearity; by step 1.1 there is a unique $X\in {\mathfrak{g}}_{\mathbb{R}}$ with $D=\mathrm{ad}X$, and $\rho =\exp (\mathrm{ad}X)$ lies in the subgroup generated by the automorphisms $\exp (\mathrm{ad}W)$, $W\in {\mathfrak{g}}_{\mathbb{R}}=\mathfrak{g}$.

The powers of $\rho$ satisfy ${\rho }^r\tau =\tau {\rho }^{-r}$ for all real $r$: from $\rho \tau ={\omega }^2\tau =\omega (\tau {\omega }^{-1})=(\omega \tau ){\omega }^{-1}=\tau {\omega }^{-1}{\omega }^{-1}=\tau {\rho }^{-1}$ one gets $\rho \tau X={\lambda }^{-1}\tau X$ on an eigenvector of eigenvalue $\lambda$, and iteration gives the claim. Put $\phi :={\rho }^{1\mathrm{/}4}$, an automorphism of ${\mathfrak{g}}_{\mathbb{R}}$ generated by the $\exp (\mathrm{ad}W)$. Then $\phi \tau {\phi }^{-1}\sigma ={\rho }^{1\mathrm{/}4}\tau {\rho }^{-1\mathrm{/}4}\sigma ={\rho }^{1\mathrm{/}2}\tau \sigma ={\rho }^{1\mathrm{/}2}{\omega }^{-1}={\rho }^{-1\mathrm{/}2}\omega =\omega {\rho }^{-1\mathrm{/}2}=\sigma \tau {\rho }^{-1\mathrm{/}2}=\sigma {\rho }^{1\mathrm{/}4}\tau {\rho }^{-1\mathrm{/}4}=\sigma \phi \tau {\phi }^{-1},$ using ${\rho }^{1\mathrm{/}2}{\omega }^{-1}={\rho }^{-1\mathrm{/}2}\omega$, the commutation of $\omega$ with powers of $\rho$, and $\tau {\rho }^{-1\mathrm{/}4}={\rho }^{1\mathrm{/}4}\tau$. Hence the involution $\psi :=\phi \tau {\phi }^{-1}$ commutes with $\sigma$.

$\psi$ is again a Cartan involution of ${\mathfrak{g}}_{\mathbb{R}}$ with $B_{\psi }(\phi Z,\phi W)=B_{\tau }(Z,W)$ positive definite, because $\phi$ is an automorphism and $\tau$ is a Cartan involution by step 2.1. Since $\psi$ commutes with $\sigma$ and $\sigma$ has fixed locus ${\mathfrak{g}}_0$ by [L1], $\psi$ preserves ${\mathfrak{g}}_0={\mathfrak{g}}^{\sigma }$: for $X\in {\mathfrak{g}}_0$ one has $\sigma (\psi X)=\psi (\sigma X)=\psi X$.

Define ${\theta }_0:=\psi {\mathrm{\mid }}_{{\mathfrak{g}}_0}\text{\hspace{0.17em}⁣}:{\mathfrak{g}}_0\to {\mathfrak{g}}_0$, a real-linear map. It is an involution, since ${\psi }^2=\mathrm{i}\mathrm{d}$, and an automorphism of ${\mathfrak{g}}_0$, since $\psi$ is an automorphism of ${\mathfrak{g}}_{\mathbb{R}}$ preserving ${\mathfrak{g}}_0$.

The form $(B_0{)}_{{\theta }_0}(X,Y)=-B_0(X,{\theta }_{0}Y)$ is positive definite. Indeed, for $X,Y\in {\mathfrak{g}}_0$ one has $B_{\mathbb{R}}(X,\psi Y)=2\mathrm{Re}B(X,\psi Y)=2B(X,\psi Y)=2B_0(X,{\theta }_{0}Y)$ by steps 1.1, 1.2 and the reality of $B$ on ${\mathfrak{g}}_0$, so $(B_0{)}_{{\theta }_0}(X,Y)=\frac{1}{2}{B}_{\psi }(X,Y)$; the form $B_{\psi }$ is positive definite on all of ${\mathfrak{g}}_{\mathbb{R}}$ by step 7.1, hence its restriction is positive definite. Therefore ${\theta }_0$ is a Cartan involution of ${\mathfrak{g}}_0$, and the theorem follows.