Exercise 15, § 2

Assume that $k$ is infinite. Let $\mathfrak{g}$ be a Lie algebra, $\mathscr{C}_{\infty}\mathfrak{g}$ the union of the ascending central series of $\mathfrak{g}$, and $x$ an element of $\mathfrak{g}$. Prove the equivalence of the following properties:

(i) $x$ belongs to every Cartan subalgebra of $\mathfrak{g}$.

(ii) $x\in \mathfrak{g}^0(y)$ for all $y\in \mathfrak{g}$ (i.e. $x\in \mathfrak{g}^0(\mathfrak{g})$).

(iii) $x\in \mathscr{C}_{\infty}\mathfrak{g}$.

(The implications (iii) $=\Rightarrow$(ii) $=\Rightarrow$(i) are immediate. To prove that (i) $=\Rightarrow$(ii), remark that (i) is equivalent to saying that $x\in \mathfrak{g}^0(y)$ for every regular element $y$ in $\mathfrak{g}$, and use the fact that the regular elements are dense in $\mathfrak{g}$ in the Zariski topology. To prove that (ii) $=\Rightarrow$ (iii), observe that $\mathfrak{n}=\mathfrak{g}^0(\mathfrak{g})$ is stable under $\mathfrak{g}($§1, Exerc. 5) and apply Engel’s theorem to the $\mathfrak{g}$-module $\mathfrak{n}$; deduce that $\mathfrak{n}$ is contained in $\mathscr{L}_{\infty}\mathfrak{g}.)$