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Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a Levi $k$-subgroup of $G$.

Does one have $\mathscr Z_{G}(S)(k) = \mathscr Z_{G(k)}(S(k))$ for the groups of $k$-rational points?

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    $\begingroup$ Maybe it's over-kill, but do you know uni-rationality of reductive groups? Hence of $S$, so its rational points are (Zariski) dense, etc? What's your context? $\endgroup$
    – paul garrett
    May 8, 2022 at 2:28
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    $\begingroup$ As @paulgarrett mentions, this is often but not always true. For example, if $G = \operatorname{SL}_2$ and $k = \mathbb F_2$, then it fails. $\endgroup$
    – LSpice
    May 8, 2022 at 3:25

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