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:
https://mathoverflow.net/questions/307465/connectedness-of-centralizers-of-semisimple-lie-algebra-elements