Let $G$ be a connected, reductive group over a field $k$. Let $A_G$ be the split component of $G$. If necessary, assume $k$ is perfect. Let $g \in G(k)$ be a semisimple element. Then the connected centralizer $Z_G(g)^0$ is defined over $k$. It is reductive (Borel, LAG, 13.19). Its Lie algebra is

$$\{ X \in \mathfrak g : \operatorname{Ad}(g)X = X\}$$

I'm trying to understand why the following statement is true:

$A_G$ is the maximal $k$-split torus of $Z_G(g)^0$ if and only if $g$ is not contained in any proper parabolic $k$-subgroups of $G$.

This is claimed in James Arthur's book, *An Introduction to the Trace Formula*, $\S 10$.

I might have a better idea of how to go about this if I had a handle on how to describe the parabolic $k$-subgroups of $Z_G(g)^0$. For example, the claim that $A_G$ is a maximal $k$-split torus of $Z_G(g)^0$ implies that $Z_G(g)^0$ cannot have any proper parabolic $k$-subgroups.