Definition 2

The ideal $n$ of Proposition 4 is called the largest nilpotency ideal for the $g$-module $M$ or the largest nilpotency ideal of the corresponding representation.

Clearly $n$ contains the kernel of this representation. It equals it when $M$ is semi-simple (Proposition 4 (c)), but not in general. It should be noted that an element of $x$ of $g$ such that $x_M$ is nilpotent does not necessarily belong to $n$.

We also note that a particular case of Lemma 3 immediately gives the following result: