Corollary

Let $\mathscr{C}_{\infty}\mathfrak{g}$ be the union of the ascending central series of the Lie algebra $\mathfrak{g}$ (Chap. I, §1, no. 6). The Cartan subalgebras of $\mathfrak{g}$ are the inverse images of the Cartan subalgebras of $\mathfrak{g}/\mathscr{C}_{\infty}\mathfrak{g}$.

Indeed, the centre of $\mathfrak{g}/\mathscr{C}_i\mathfrak{g}$ is $\mathscr{C}_{i+1}\mathfrak{g}/\mathscr{C}_i\mathfrak{g}$, and the corollary follows immediately from Prop. 5 by induction.