Corollary 3

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie algebra $\mathfrak{g}$, and let $\mathscr{C}^n\mathfrak{g}(n\geq 1)$ be a term of the descending central series of $\mathfrak{g}$ (Chap. I, §1, no. 5). Then $\mathfrak{g}=\mathfrak{h}+\mathscr{C}^n\mathfrak{g}$.

Indeed, Corollary 2 shows that the image of $\mathfrak{h}$ in $\mathfrak{g}/\mathscr{C}^n\mathfrak{g}$ is a Cartan subalgebra of $\mathfrak{g}/\mathscr{C}^n\mathfrak{g}$, hence is equal to $\mathfrak{g}/\mathscr{C}^n\mathfrak{g}$ since $\mathfrak{g}/\mathscr{C}^n\mathfrak{g}$ is nilpotent (Example 1).