Let $m,n,k$ be nonnegative integers with $m+k=n$. Let $G=\mathrm O(m,k)\subseteq \mathrm{GL}_n(\mathbb{R})$ be the subgroup of matrices preserving the standart non-degenerate symmetric billinear form of signature $(m,k)$ and $K\subseteq G$ be a maximal compact subgroup. I have seen people state without proof that $K=\mathrm O(m)\times \mathrm O(k)$. Trying to prove this fact I understand the following:
(i) By choosing a $K$ - invariant inner product we can see that $K=G\cap H$ when $H$ is some subgroup of $\mathrm{GL}_n(\mathbb{R})$ conjugate in $\mathrm{GL}_n(\mathbb{R})$ to $O(n)$.
(ii) If this $H$ would be just $\mathrm O(n)$ then we would get the above.
My questions are:
(1) How can I understand whether or not we can choose the $K$ - invariant inner product so that $H=\mathrm O(n)$?
(2) Can different choices of such an $H$ (i.e.: a subgroup of $\mathrm{GL}_n(\mathbb{R})$ conjugate inside $\mathrm{GL}_n(\mathbb{R})$ to $\mathrm O(n)$) really lead to different intersections $G\cap H$? (this question has two versions: as subgroups of $G$ up to conjugation, and as abstract lie groups. I'm interested in both.)
(3) Noticing that (i) remains valid for any subgroup $G\subseteq \mathrm{GL}_n(\mathbb{R})$, what can be said about questions (1) and (2) for an arbitrary (say lie) subgroup $G\subseteq \mathrm{GL}_n(\mathbb{R})$?
Anything that can be said about this sort of stuff or a reference dealing with it would be appreciated.