Corollary 2

Let g be a Lie algebra and h an ideal of g. Suppose that $g/h$ is nilpotent and that, for all $x \in g$, the restriction of $\mathrm{ad}\ x$ to h is nilpotent. Then g is nilpotent.

Let $x \in g$. As $g/h$ is nilpotent, there exists an integer $k$ such that $(\mathrm{ad}\ x)^k(g) \subset h$. By hypothesis there exists an integer $k'$ such that $(\mathrm{ad}\ x)^{k'}(h) = \{0\}$. Hence $(\mathrm{ad}\ x)^{k+k'}(g) = \{0\}$. Corollary 2 is hence a consequence of Corollary 1.