Definition 1

Let $\mathfrak{q}$ be a Lie subalgebra of the semi-simple Lie algebra $\mathfrak{g}$. Then $\mathfrak{q}$ is said to be decomposable in $\mathfrak{g}$ if, for all $x\in \mathfrak{q}$, the semi-simple and nilpotent components of $x$ in $\mathfrak{g}$ belong to $\mathfrak{q}$. Denote by $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{q})$ the set of elements $x$ of the radical of $\mathfrak{q}$ such that ad$_{\mathfrak{g}}x$ is nilpotent.

Let $\rho$ be an injective representation of $\mathfrak{g}$ on a finite dimensional vector space V. We know (Chap. I, §6, no. 3, Th. 3) that an element $x$ of $\mathfrak{g}$ is semi-simple (resp. nilpotent) if and only if the endomorphism $\rho (x)$ of V is semi-simple (resp. nilpotent). It follows immediately that the algebra $\mathfrak{q}$ is decomposable in $\mathfrak{g}$ if and only if $\rho (\mathfrak{q})$ is a decomposable subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ in the sense of Definition 1 of Chap. VII, §5, no. 1. With the notations of Chap. VII, §5, no. 3, we also have