While zyxel has provided a concise answer and reference, it's worth filling in more details about the original source of this kind of result. Unfortunately, it wasn't clearly articulated in textbooks before Digne-Michel (who were especially interested in the structure of groups over finite fields following the work of Deligne and Lusztig).

1) First, I'd emphasize that *simply connected* only makes sense here for a semisimple group, and in any case doesn't influence the answer to your question. The basic object of study is a connected reductive group $G$, but one might as well focus here on the connected semisimple derived group when it has positive dimension. This is where the combinatorics of root systems matters.

2) In the question, $G$ can be taken to be defined (and automatically split) over any algebraically closed field $k$ of arbitrary characteristic. But the detailed structure theory involving centralizers of tori and parabolics was first undertaken over a general field of definition by Borel and Tits in their foundational 1965 paper
here. The relevant material (applicable whenever $G$ is $k$-isotropic) is contained in sections 3 and 4, with a fairly explicit general statement in Theorem 4.15. What Digne and Michel do is essentially extracted from this source.

3) The ideas of Borel-Tits were intentionally coordinated at the time with Chapter VI of Bourbaki's treatise *Groupes et algebres de Lie*, treating root systems axiomatically. This was only published later in 1968, so references in Borel-Tits are numbered tentatively. In the published Bourbaki volume the crucial results are in section 1.7, especially Prop. 23, 24.

4) Its easy to get lost in the details, especially when working over arbitary $k$, so it may be helpful to outline briefly what goes into the basic proof that the centralizer $H:=C_G(S)$ of an arbitrary torus $S$ is a Levi factor of some parabolic subgroup of $G$. First, it follows readily from the Borel-Chevalley structure theory that $H$ is *connected* and *reductive* (and this is contained in standard textbooks). Taking an arbitrary basis for the root system of the derived group of $H$, the key point is to embed this basis in a basis of the full root system of $G$ relative to a maximal torus $T$ containing $S$. (This is where Bourbaki's abstract arguments come into play.) Then the Borel-Tits study of parabolics leads to a parabolic subgroup in which $H$ is a Levi factor. (Of course, this will turn out to be one of the standard parabolics of $G$ relative to $T$.)

singlesemisimple element is connected. $\endgroup$ – LSpice Sep 16 '19 at 21:07