Proposition 4

Let $\mathfrak{a}$ be a subalgebra of $\mathfrak{g}$ containing $\mathfrak{h},x$ an element of $\mathfrak{a},s$ and $n$ its semi-simple and nilpotent components. Then $s\in \mathfrak{a}$ and $n\in \mathfrak{a}$.

We have (ad $x$)$\mathfrak{a}\subset \mathfrak{a}$, so (ad $s$)$\mathfrak{a}\subset \mathfrak{a}$ and (ad $n$)$\mathfrak{a}\subset \mathfrak{a}$. Since $\mathfrak{a}$ is its own normalizer in $\mathfrak{g}$ (Chap. VII, §2, no. 1, Cor. 4 of Prop. $4$)$,s\in \mathfrak{a}$ and $n\in \mathfrak{a}$.