Theorem 1

Let $\mathfrak{a}$ be a complex semi-simple Lie algebra.

a$)\mathfrak{a}$ has compact (resp. splittable) real forms.

b) The group Int($\mathfrak{a}$) operates transitively on the set of compact (resp. splittable) real forms of $\mathfrak{a}$.

Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{a}$. Then $(\mathfrak{a},\mathfrak{h})$ is split (Chap. VIII, §2, no. 1, Remark 2), and has a Chevalley system $(X_{\alpha})$ (Chap. VIII, §4, no. 4, Cor. of Prop. 5). Part a) now follows from Prop. 2. Let $\mathfrak{g}$ be a compact real form of $\mathfrak{a}$; we show that there exists $v\in$ Int($\mathfrak{a}$) such that $v(\mathfrak{a}_u) =\mathfrak{g}$. Let $\mathfrak{t}$ be a Cartan subalgebra of $\mathfrak{g}$; then $\mathfrak{t}_{(\mathbf{C})}$ is a Cartan subalgebra of $\mathfrak{a}$; since Int($\mathfrak{a}$) operates transitively on the set of Cartan subalgebras of $\mathfrak{a}$ (Chap. VII, §3, no. 2, Th. 1), we are reduced to the case in which $\mathfrak{t}_{(\mathbf{C})}=\mathfrak{h}$. Since $\mathfrak{g}$ is a compact form, the eigenvalues of the endomorphisms ad $h$, for $h\in \mathfrak{t}$, are purely imaginary (§1, no. 3, Prop. 1), so the roots $\alpha \in R$ map $\mathfrak{t}$ to $i\mathbf{R}$; this implies that $\mathfrak{t}=i\mathfrak{h}_0$. Then, by Prop. 3 (no. 2), there exists $v\in$ Int($\mathfrak{a}$) such that $v(\mathfrak{a}_u) =\mathfrak{g}$, hence b) in the case of compact forms. Finally, let $\mathfrak{m}_1$ and $\mathfrak{m}_2$ be two splittable real forms of $\mathfrak{a}$. There exist framings $(\mathfrak{m}_1,\mathfrak{h}_1,B_1,(X_{\alpha}^1))$ and $(\mathfrak{m}_2,\mathfrak{h}_2,B_2,(X_{\alpha}^2))$ (Chap. VIII, §4, no. 1). These extend in an obvious way to bases $e_1$ and $e_2$ of $\mathfrak{a}$. An automorphism of $\mathfrak{a}$ that maps $e_1$ to $e_2$ maps $\mathfrak{m}_1$ to $\mathfrak{m}_2$; thus, it suffices to apply Prop. 5 of Chap. VIII, §5, no. 3, to obtain the existence of an element $u$ of Aut$_0(\mathfrak{a}) =$ Int($\mathfrak{a}$) such that $u(\mathfrak{m}_1) =\mathfrak{m}_2$.