Corollary 1

The Cartan subalgebras of $\mathfrak{g}$ are the centralizers of the regular semi-simple elements of $\mathfrak{g}$.

If $x\in \mathfrak{g}$ is regular, $\mathfrak{g}^0(x)$ is a Cartan subalgebra of $\mathfrak{g}($§2, no. 3, Th. 1 (i)); moreover, if $x$ is semi-simple $\mathfrak{g}^0(x)$ is the centralizer of $x$ in $\mathfrak{g}$. Conversely, let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. There exists $\mathfrak{t}\in \mathscr{T}_1$ such that $\mathfrak{h}=\psi (\mathfrak{t})$. By §1, no. 2, Prop. 7, there exists $x\in \mathfrak{t}$ such that $\mathfrak{h}=\mathfrak{g}^0(x)$; since $x\in \mathfrak{t}$, $\mathfrak{g}^0(x) =\mathfrak{g}_0(x)$. By §3, no. 3, Th. 2 (ii), $x$ is regular.