Lemma 1

Let $g = g' + h$ be a Lie algebra which is the sum of an ideal $g'$ and a subalgebra $h$. Let $\sigma$ be a finite-dimensional representation of $g$. Suppose that $\sigma(x)$ is nilpotent for all $x \in g'$ and all $x \in h$. Then $\sigma(x)$ is nilpotent for all $x \in g$.

Passing to the quotient by the kernel of $\sigma$, $\sigma$ may be assumed to be faithful. Then $g'$ and $h$ are nilpotent and hence $g$, which is an extension of a quotient of $h$ by $g'$, is solvable. Then $h$ and $g'$ are contained in the largest nilpotency ideal of $\sigma$ (§ 5, no. 3, Corollary 6 to Theorem 1).

For an improvement of Theorem 1, cf. Exercise 4.