Exercise 8, § 3

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{a}$ an ideal of $\mathfrak{g},\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ and $\mathscr{C}_{\infty}\mathfrak{g}$ the union of the ascending central series of $\mathfrak{g}$. Show that $\mathfrak{a}\subset \mathfrak{h}$ implies $\mathfrak{a}\subset \mathscr{C}_{\infty}\mathfrak{g}$ (in other words $\mathscr{C}_{\infty}\mathfrak{g}$ is the largest ideal of $\mathfrak{g}$ contained in $\mathfrak{h}$). (Reduce to the case where $k$ is algebraically closed and remark that $\mathfrak{a}$ is stable under every elementary automorphism of $\mathfrak{g}$; the relation $\mathfrak{a}\subset \mathfrak{h}$ then implies that $\mathfrak{a}$ is contained in every Cartan subalgebra of $\mathfrak{g}$; conclude by means of Exerc. 15 of §2.)