Orthogonal group is finite-dominating in general linear group over any real-closed field

Definition
Let $$k$$ be a real-closed field, for instance, $$k = \R$$, and $$n$$ be any natural number. Then, any finite subgroup of the general linear group $$GL(n,k)$$ is conjugate in $$GL(n,k)$$ to a subgroup of the orthogonal group $$O(n,k)$$.

Related facts

 * General linear group is finite-dominating in general affine group over characteristic zero