Let $\mathfrak{g}$ be a complex simple Lie algebra and $x \in \mathfrak{g}$ be a regular element, i.e. its centralizer is of minimal dimension.

Consider the adjoint action of the adjoint group $G$ (with trivial center) on its Lie algebra $\mathfrak{g}$.

Is it true that the stabilizer of $x$ in $G$ is always connected?

If $x$ is semisimple, then it is known that the stabilizer of $x$ in the simply connected group is connected (see the book of Collingwood-McGovern, Theorem 2.3.3), and so by projection, also in the adjoint group.

If $x$ is regular and nilpotent (also called principal nilpotent), then one can check in the lists of all Lie algebras in the book of Collingwood-McGovern to check that the stabilizer is connected.

But what about a general regular element? And if it is true, is there a simple proof (without using the classification of simple Lie algebras)?