Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
where ${\mathfrak g}={\rm Lie}\ G$.


I am looking for a *reference* to a proof of the following assertion:



> 
> **Proposition.** *For any* semisimple *element $X\in{\mathfrak g}$, 
>  its stabilizer ${\rm Stab}\_G(X)\subset G$ with respect to the adjoint representation
>  is connected.*
> 
> 
> 


I think that I can prove the proposition; see my proof below.


*Proof.* Our semisimple element $X$ is contained in a Cartan subalgebra ${\mathfrak t}$ of ${\mathfrak g}$, which is the Lie algebra of a maximal torus $T$ of $G$. Then ${\mathfrak t}$ is an algebraic subalgebra of ${\mathfrak g}$.
Let ${\langle X\rangle\_{\rm alg}}\subset {\mathfrak g}$ denote the smallest algebraic subalgebra of ${\mathfrak g}$ containing $X$;
then ${\langle X\rangle\_{\rm alg}}\subseteq {\mathfrak t}$. It follows that ${\langle X\rangle\_{\rm alg}}={\rm Lie}\ S$ for some subtorus $S\subseteq T$.
Now (in characteristic 0) we have
$${\rm Stab}\_G(X)=\bigcap\_{Y\in {\langle X\rangle\_{\rm alg}}} {\rm Stab}\_G(Y)=C\_G(S),$$
where $C\_G(S)$ denotes the centralizer of the torus $S$ in $G$.
By Theorem 22.3 of Humphreys' book "Linear Algebraic Groups", 
$C\_G(S)$ is connected, as required.


**Edit:** A similar argument shows that for any commutative subalgebra ${\mathfrak a}\subset {\mathfrak g}$ consisting of semisimple elements, its centralizer in $G$
$$ C\_G({\mathfrak a}):=\bigcap\_{X\in {\mathfrak a}} {\rm Stab}\_G(X)$$
is connected (because the "algebraic closure" $\langle {\mathfrak a}\rangle\_{\rm alg}$ of $\mathfrak a$ is the Lie algebra of some torus $S\subset G$).



Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request
A reference: [Steinberg, Torsion in reductive groups, 
Advances in Math. 15 (1975), 63–92](https://www.sciencedirect.com/science/article/pii/0001870875901255?via%3Dihub), Corollary 3.11. 


In positive characteristic $p$: see *loc. cit.*, Theorem 3.14. It says that if (and only if) $p$ is not a torsion prime for $G$, then $C\_G({\mathfrak a})$ is connected for any commutative subalgebra ${\mathfrak a}\subset {\mathfrak g}$ consisting of semisimple elements.


See also:
[Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group](https://mathoverflow.net/questions/307465/connectedness-of-centralizers-of-semisimple-lie-algebra-elements)


