Proposition 3

Let $\mathfrak{a}$ be a semi-simple subalgebra of $\mathfrak{g}$ stable under ad($\mathfrak{h}$) and let P be the subset of R such that $\mathfrak{a}=\mathfrak{h}_P+\mathfrak{g}^P$.

(i) $\mathfrak{h}_P$ is a splitting Cartan subalgebra of $\mathfrak{a}$.

(ii) The root system of $(\mathfrak{a},\mathfrak{h}_P)$ is the set of restrictions to $\mathfrak{h}_P$ of elements of P.

Since $\mathfrak{h}_P$ is stable under ad $\mathfrak{h}$, its normalizer in $\mathfrak{a}$ is stable under ad $\mathfrak{h}$, and hence is of the form $\mathfrak{h}_P+\mathfrak{g}^Q$ where $Q\subset P$ (Lemma 1). If $\alpha \in Q$,

which is absurd. Thus $Q =\emptyset$ and $\mathfrak{h}_P$ is its own normalizer in $\mathfrak{a}$. This proves that $\mathfrak{h}_P$ is a Cartan subalgebra of $\mathfrak{a}$. If $x\in \mathfrak{h}_P$, ad$_{\mathfrak{g}}x$, and a fortiori ad$_{\mathfrak{a}}x$, are triangularizable. Thus (i) is proved, and (ii) is clear.

By Prop. 1 (i), the subalgebras of $\mathfrak{g}$ containing $\mathfrak{h}$ are the sets $\mathfrak{h}+\mathfrak{g}^P$ where P is a closed subset of R. By Chap. VII, §3, Prop. 3, every Cartan subalgebra of $\mathfrak{h}+\mathfrak{g}^P$ is a Cartan subalgebra of $\mathfrak{g}$.