Theorem 2

Let $\mathfrak{g}$ be a Lie algebra.

(i) The Cartan subalgebras of $\mathfrak{g}$ are all of the same dimension, namely rk($\mathfrak{g}$), and the same nilpotency class.

(ii) An element $x\in \mathfrak{g}$ is regular if and only if $\mathfrak{g}^0(x)$ is a Cartan subalgebra of $\mathfrak{g}$; every Cartan subalgebra is obtained in this way.

To prove (i), we can assume that $k$ is algebraically closed (cf. §2, Prop. 3 and Prop. 6), in which case it follows from Th. 1 of no. 2. Assertion (ii) follows from (i) and §2, Th. 1 (i) and (iv).