Lemma 2

Let g be a Lie algebra, a an ideal of g and M a simple g-module. If, for all $x \in a,\ x_M$ is nilpotent, then $x_M = 0$ for all $x \in a$.

Let N be the subspace of M consisting of the $m \in M$ such that $x_M.m = 0$ for all $x \in a$. By Theorem 1, $N \neq \{0\}$. On the other hand, for all $y \in g$, N is stable under $y_M$ (§ 3, no. 5, Proposition 5). Hence $N = M$, which proves the lemma.