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Let $G$ be any algebraic subgroup of $\mathrm{GL}_n$ over an algebraically closed field of any characteristic.

If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be closed in $G$?

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No---it's always closed. See 18.2 in Humphreys, Linear Algebraic Groups.

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  • $\begingroup$ Is it still true when the field is not algebraically closed? $\endgroup$
    – Michiel Van Couwenberghe
    Commented Jan 31, 2018 at 12:52

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