Corollary 5

Let $\mathfrak{g}$ be a Lie algebra, E a subset of $\mathfrak{g}$. Let E operate on $\mathfrak{g}$ by the adjoint representation. Then E is a Cartan subalgebra of $\mathfrak{g}$ if and only if $E =\mathfrak{g}^0(E)$.

The condition is necessary (Prop. 4). Assume now that $E =\mathfrak{g}^0(E)$. By Prop. 2 (ii) of §1, no. 1, E is then a subalgebra of $\mathfrak{g}$. If $x\in E$, ad$_Ex$ is nilpotent since $E\subset \mathfrak{g}^0(E)$; hence the algebra E is nilpotent. But then E is a Cartan subalgebra by Prop. 4.