Proposition 4

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{h}$ a nilpotent subalgebra of $\mathfrak{g}$. Then $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ if and only if $\mathfrak{g}^0(\mathfrak{h}) =\mathfrak{h}$.

If $\mathfrak{g}^0(\mathfrak{h}) =\mathfrak{h},\mathfrak{h}$ is its own normalizer (§1, Prop. 10 (i)), so $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. Assume that $\mathfrak{g}^0(\mathfrak{h})\not=\mathfrak{h}$. Consider the representation of $\mathfrak{h}$ on $\mathfrak{g}^0(\mathfrak{h})/\mathfrak{h}$ obtained from the adjoint representation by passage to the quotient. By applying Engel’s theorem (Chap. I, §4, no. 2, Th. 1), we see that there exists $x\in \mathfrak{g}^0(\mathfrak{h})$ such that $x \notin \mathfrak{h}$ and $[\mathfrak{h}, x]\subset \mathfrak{h}$; then $x$ belongs to the normalizer of $\mathfrak{h}$ in $\mathfrak{g}$, so $\mathfrak{h}$ is not a Cartan subalgebra of $\mathfrak{g}$.