Corollary 2

Let $\mathfrak{n}$ be a subalgebra of $\mathfrak{g}$ consisting of nilpotent elements. There exists a parabolic subalgebra $\mathfrak{q}$ of $\mathfrak{g}$ with the following properties:

(i) $\mathfrak{n}\subset \mathfrak{n}_{\mathfrak{g}}(\mathfrak{q})$;

(ii) the normalizer of $\mathfrak{n}$ in $\mathfrak{g}$ is contained in $\mathfrak{q}$;

(iii) every automorphism of $\mathfrak{g}$ leaving $\mathfrak{n}$ invariant leaves $\mathfrak{q}$ invariant.

If $\mathfrak{g}$ is splittable, $\mathfrak{n}$ is contained in a Borel subalgebra of $\mathfrak{g}$.

Let $\mathfrak{q}_1$ be the normalizer of $\mathfrak{n}$ in $\mathfrak{g}$. This is a decomposable subalgebra of $\mathfrak{g}$. Let $\mathfrak{n}_1=\mathfrak{n}_{\mathfrak{g}}(\mathfrak{q}_1)$. Define inductively $\mathfrak{q}_i$ to be the normalizer of $\mathfrak{n}_{i-1}$ in $\mathfrak{g}$, and $\mathfrak{n}_i$ to be equal to $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{q}_i)$. The sequences $(\mathfrak{n},\mathfrak{n}_1,\mathfrak{n}_2, . . .)$ and $(\mathfrak{q}_1,\mathfrak{q}_2, . . .)$ are increasing. There exists $j$ such that $\mathfrak{q}_j=\mathfrak{q}_{j+1}$, in other words $\mathfrak{q}_j$ is the normalizer of $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{q}_j)$ in $\mathfrak{g}$. Thus $\mathfrak{q}_j$ is parabolic (Th. 1). We have $\mathfrak{n}\subset \mathfrak{n}_j=$ $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{q}_j)$, and $\mathfrak{q}_1\subset \mathfrak{q}_j$; every automorphism of $\mathfrak{g}$ leaving $\mathfrak{n}$ invariant evidently leaves $\mathfrak{n}_1,\mathfrak{n}_2, . .$. and $\mathfrak{q}_1,\mathfrak{q}_2, . .$. invariant. If $\mathfrak{g}$ is splittable, $\mathfrak{q}_j$ contains a Borel subalgebra $\mathfrak{b}$, and consequently (§3, no. 4, Prop. 13), we have $\mathfrak{b}\supset \mathfrak{n}_{\mathfrak{g}}(\mathfrak{q}_j)\supset \mathfrak{n}$.