Cartan involution of a real semisimple Lie algebra

Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with Killing form $B$ (Killing form). A Cartan involution of ${\mathfrak{g}}_0$ is a Lie-algebra automorphism $\theta \text{\hspace{0.17em}⁣}:{\mathfrak{g}}_0\to {\mathfrak{g}}_0$ with ${\theta }^2=\mathrm{i}\mathrm{d}$ for which the symmetric bilinear form

$B_{\theta }(X,Y):=-B(X,\theta Y)$

is positive definite on ${\mathfrak{g}}_0$. The associated Cartan decomposition is the eigenspace decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ into the $+1$- and $-1$-eigenspaces of $\theta$ (Cartan decomposition of a real semisimple Lie algebra). Since $\theta$ is an involution, ${\mathfrak{k}}_0,{\mathfrak{p}}_0$ are the fixed and anti-fixed subspaces respectively, and $B_{\theta }$ being positive definite makes the decomposition orthogonal for $B$ with $B$ negative definite on ${\mathfrak{k}}_0$ and positive definite on ${\mathfrak{p}}_0$ (Bracket relations and Killing signs in a Cartan decomposition). Under the Axiom of Choice (The Axiom of Choice), existence of Cartan involutions is proved in Existence of a Cartan involution.