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

Question: Why is this group well defined? Although it seems rather natural to construct, namely we can embed $G$ in some $\GL_n$ and then take as $G(\mathcal{O}_K):=G(K) \cap \GL_n(\mathcal{O}_K)$, later exists obviously "canonically. But here is on the other hand 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, right?

Or is it nevertheless "canonical" by some additional argument?