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Suppose some maximal torus $T$ of $G$ is $C_G(T)$, then the set of semisimple elements for which $C_G(s)$ is a torus contains a nonempty open set. Such element are called regular semisimple.

I want to know how to prove this claim and want to find some good examples for some common algebraic groups such as $\mathrm{SL}(n,K)$.

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1 Answer 1

(I will assume you are still interested in getting an answer to this problem.)

By Lemma 12.2 of Borel's Linear Algebraic Groups, being regular is equivalent to being fixed by no roots. Therefore, the set of regular elements is a finite intersection of open sets, and so it is open.

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