Proposition 6

Let $g$ be a Lie algebra and $E$ the associative subalgebra of $\mathscr{L}(g)$ generated by 1 and the $\operatorname{ad}_g x$ ($x \in g$). Let $R$ be the Jacobson radical of $E$. (a) The set $n$ of $y \in g$ such that $\operatorname{ad}_g y \in R$ is the largest nilpotent ideal of $g$. (b) It is orthogonal to $g$ under the Killing form.

It should be noted that $g/n$ can have non-zero nilpotent ideals.