Corollary 7

Let g be a Lie algebra and r its radical. The following four sets are identical: (a) the largest nilpotent ideal of g; (b) the largest nilpotent ideal of r; (c) the set of $x \in r$ such that $\mathrm{ad}_g x$ is nilpotent; (d) the set of $x \in r$ such that $\mathrm{ad}_r x$ is nilpotent.

Let these sets be denoted by $a, b, c, d$. The inclusions $a \subset b \subset d \subset c$ are clear. $c \subset a$ by Corollary 6 applied to the adjoint representation of g.