$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group, $K$ be a non-Archimedean local field and $\mathcal{O}_K$ be its ring of integers.

In context of Satake theory one considers as important object for further constructions the group $G(\mathcal{O}_K)$ of integer points of $G$.

Question: Why is this group well defined at all? Although it seems rather natural to construct, namely we can embed $G$ in some $\GL_n$ by definition of algebraic group and then set $G(\mathcal{O}_K):=G(K) \cap \GL_n(\mathcal{O}_K)$, latter exists obviously "canonically". But here is of course an explicit choice of $\GL_n$ involved where $G$ in going to be embedded, so $G(\mathcal{O}_K)$ seems to be dependent on an explicit embedding $ G \subset \GL_n$, isn' t it?

Or is it nevertheless $G(\mathcal{O}_K)$ "canonical" by some additional argument?