A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, Corollary 3.11.

In positive characteristic: loc. cit, Theorem 3.14.

See also: Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, 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

A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, Corollary 3.11.

In positive characteristic: loc. cit, Theorem 3.14.

See also: Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, Corollary 3.11.

In positive characteristic: loc. cit, Theorem 3.14.

See also: Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group