Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the intersections $H\cap \text{GL}(n, k)$ and $H\cap \text{GL}(n, \mathcal{O}_k)$.

Let $H$ be a semisimple algebraic group defined over $k$. I need to show that $H(k)$ commensurates $H(\mathcal{O}_k)$, i.e. that for every $\gamma\in H(k)$ the subgroup $\gamma H(\mathcal{O}_k)\gamma^{-1}$ is commensurable to $H(\mathcal{O}_k)$.

I do not have any idea of how to proceed. The only observation I have done is that since the commensurator of a subgroup is group, and the constant matrices commensurate $H(\mathcal{O}_k)$, I can suppose that $\gamma$ has entries in $\mathcal{O}_k$, but in this way I probably lose the information of $\gamma$ being an element of $H$.

Ifyou fail to secure responses in one site and you think you might get them in the other, then indicate in your post that it is a cross-post, to avoid possible duplication of effort. $\endgroup$$H$ is semisimple": this doesn't make sense for an arbitrary subgroup, but for an algebraic subgroup, or a closed one. $\endgroup$2more comments