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char k=0 is necessary for the semi-simplicity of the elements of $G(k)$.
Venkataramana
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The answer is always. If $G$ is an anisotropic group over a non-archimedean local field $k$, then $G(k)$ is compact. The elements of $G(k)$ are semi-simple (if $char k =0$) and hence their conjugacy classes are closed in the Zariski topology and therefore also in the $k$-topology.

Venkataramana
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