Corollary 6

Let $\varphi$ be a finite-dimensional representation of a Lie algebra $g$. Let r be the radical of g. Every element $x \in r$ such that $\rho(x)$ is nilpotent belongs to the largest nilpotency ideal $n$ of $\rho$.

Let V be the space of $\rho$; let $(V_i)_{0 \leq i \leq r}$ be a Jordan-Hölder series for the r-module structure on V and let $\rho_i$ be the representation of r with space $V_i / V_{i-1}$ ($1 \leq i \leq r$). If $\rho(x)$ is nilpotent, so is $\rho_i(x)$; as for all $i$ the algebra generated by $\rho_i(x)$ is a field, $\rho_i(x) = 0$. Conversely, if $\rho_i(x) = 0$ for all $i$, $\rho(x) = 0$. This shows that the set $a$ of $x \in r$ such that $\rho(x)$ is nilpotent is an ideal of r. On the other hand, $[g, a] \subset \mathscr{D}g \cap r \subset n \cap r \subset a$ and hence $a$ is an ideal of g. This proves that $a \subset n$.