Cartan subalgebras are self-normalizing

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over an algebraically closed field of characteristic $0$, and let $\mathfrak{h}\subset \mathfrak{g}$ be a Cartan subalgebra .

Lemma. The normalizer of $\mathfrak{h}$ in $\mathfrak{g}$ is $\mathfrak{h}$ itself:

Equivalently, if $X\in\mathfrak{g}$ satisfies $[X,H]\in \mathfrak{h}$ for every $H\in\mathfrak{h}$, then $X\in \mathfrak{h}$.

Context. This property ensures that $\mathfrak{h}$ is as large as possible among nilpotent subalgebras compatible with its own adjoint action: anything that stabilizes $\mathfrak{h}$ by commutators is already inside $\mathfrak{h}$. In semisimple Lie theory, self-normalizing is what makes the root decomposition relative to $\mathfrak{h}$ behave rigidly, and it underlies the definition of the Weyl group as a quotient of a group normalizer by a centralizer.