Real form of a complex Lie algebra

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. A real form of $\mathfrak{g}$ is a real Lie subalgebra ${\mathfrak{g}}_0\subseteq \mathfrak{g}$ such that the complex-linear extension

${\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}⟶\mathfrak{g},X\otimes z⟼zX,$

of the inclusion ${\mathfrak{g}}_0\hookrightarrow \mathfrak{g}$ (Complexification of a real Lie algebra) is an isomorphism of complex Lie algebras. Equivalently, ${\mathfrak{g}}_0$ is a real subspace of $\mathfrak{g}$ with ${\mathfrak{g}}_0\cap i{\mathfrak{g}}_0=0$ and $\mathfrak{g}={\mathfrak{g}}_0\oplus i{\mathfrak{g}}_0$ (a real direct sum), and ${\mathfrak{g}}_0$ is closed under the bracket. A conjugate-linear involution of $\mathfrak{g}$ is a conjugate-linear map $\sigma \text{\hspace{0.17em}⁣}:\mathfrak{g}\to \mathfrak{g}$ with $\sigma \circ \sigma ={\mathrm{i}\mathrm{d}}_{\mathfrak{g}}$ and $\sigma [Z,W]=[\sigma Z,\sigma W]$ for all $Z,W\in \mathfrak{g}$. Two real forms ${\mathfrak{g}}_0,{\mathfrak{g}}_0^{\mathrm{\prime }}$ are conjugate if ${\mathfrak{g}}_0^{\mathrm{\prime }}=u({\mathfrak{g}}_0)$ for a complex Lie-algebra automorphism $u$ of $\mathfrak{g}$, and two conjugate-linear involutions $\sigma ,{\sigma }^{\mathrm{\prime }}$ are conjugate if ${\sigma }^{\mathrm{\prime }}=u\circ \sigma \circ u^{-1}$ for such a $u$. The correspondence between real forms and conjugate-linear involutions is proved in Real forms correspond to conjugate-linear involutions.