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$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ 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?

$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ 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?

#EDIT: As @David Loeffler' s answer states, the claim without certain additional assumptions on $G$ is wrong, so I would like to additionally assume as @LSpice suggested that $G$ is moreover split, conjecturing that this may provide a sufficient assumption for existence of the group I' m looking for.

$\DeclareMathOperator\GL{GL}$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?

$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ 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?

$\DeclareMathOperator\GL{GL}$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?

$\DeclareMathOperator\GL{GL}$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?