EDIT: The question and some of the responses are out of focus, so it's worth clarifying a few of the issues here. First, the intended notion of "regular element" is unclear. In Kostant's early papers, for example *here*, only regular *semisimple* elements in semisimple Lie groups are directly considered. But later he broadened the definition. In the general Lie group framework, there is no notion of "Jordan-Chevalley decomposition", so a broader definition of regularity was given by Steinberg for a connected semisimple algebraic group $G$ *here*. An element $x \in G$ is called *regular* if its centralizer has dimension equal to the rank $\ell$ of $G$ (which is the least possible).

At the 1966 ICM in Moscow, Steinberg surveyed questions about conjugacy classes and regular elements
*here*. Among the problems he raised were (14) and (15), asking whether regular $x \in G$ is characterized by having a commutative centralizer. This and more was later confirmed by two of his students
B. Lou and
S.V. Keny, even for fields of "bad" characteristic.

For $X \in \mathfrak{g}$, the Lie algebra of $G$, there are actually two relevant centralizers: the subgroup $G_X$ of $G$ fixing $X$ under the adjoint representation Ad, and the subalgebra $\mathfrak{g}_X$ of $\mathfrak{g}$ fixing $X$ under ad. The latter always contains the Lie algebra of $G_X$ and equals it in characteristic 0. Springer refined and extended Steinberg's ideas in a 1966 paper *here*: see $\S4$ and $\S5$. He further showed in Proposition 2 *here* that when $x \in G$ is regular, the identity component of $G_x$ is commutative. The work of Lou and Keny showed in fact that $G_x$ itself is commutative, while $G_X$ is commutative for a regular element $X\in \mathfrak{g}$ provided $G$ is of *adjoint* type.

Thanks to the fact that the centralizer of a semisimple element is *reductive* (the almost-direct product of a central torus and a semisimple group), the main problems are reduced to $x \in G$ unipotent or $X \in \mathfrak{g}$ nilpotent. But $G_x$ is not necessarily connected for $x$ semisimple unless $G$ is simply connected. (An
algebraic proof due to Digne-Michel is discussed in Chapter 2 of my 1995 book on conjugacy classes; the first proof was by Springer and Steinberg.) A similar reduction to the regular nilpotent case works for $X \in \mathfrak{g}$ and $G_X$ when $G$ is simply connected.

In the case of $G_X$ for $X$ regular nilpotent, the outcome in characteristic 0 is fairly straightforward: here the centralizer is the product of the center of $G$ and a (connected!) commutative unipotent group. But for bad prime characteristics, it took some case-by-case work to describe $G_X$.

containedin the generalized $0$-weight space of $\mathrm{ad}(X)$ which is a Cartan subalgebra since $X$ is regular. $\endgroup$