We will say that $G<GL(n;\mathbb{R})$ is an orthogonal group if $G$ is abstractly isomorphic to $O(p,q)$ for some $p+q=n$. If $G\cong O(p,q)$ is an orthogonal group, is $G$ actually conjugate in $GL(n;\mathbb{R})$ to $O(p,q)$?
I think this follows for groups $G$ abstractly isomorphic to $O(n)$ as this is the maximal compact subgroup and hence unique up to conjugacy. I'm looking for suggestions on how to go about proving (or looking for counterexamples) to the indefinite cases.