Proposition 4

Let $g$ be a Lie algebra, $M$ a $g$-module of finite dimension over $K$ and $A$ the associative algebra generated by 1 and the set of $x_M$ ($x \in g$).

(a) The ideals $a$ of $g$ such that $x_M$ is nilpotent for all $x \in a$ are all contained in one of them, $n$.

(b) The ideal $n$ is the set of $x \in g$ such that $x_M$ belongs to the Jacobson radical of $A$.

(c) Let $(M_i)_{0 \leq i \leq n}$ be a Jordan-Hölder series of the $g$-module $M$; then $n$ is also the set of $x \in g$ such that $(x)_{M_i/M_{i+1}} = 0$ for all $i$.

(d) $n$ is orthogonal to $g$ with respect to the bilinear form associated with $\varphi$.

The set of $x \in g$ such that $x_M$ belongs to the Jacobson radical of $A$ is obviously an ideal of $g$. The proposition then follows immediately from Lemma 3.