Let $O(n)$ and $SO(n)$ denote the orthogonal and special orthogonal groups respectively on $\mathbb{R}^n$. In classical invariant theory, the first fundamental theorem of invariant theory states the following:
Theorem: The ring of invariant polynomials for $O(n)$ is generated by the metric. Likewise, the ring of invariant polynomials for $SO(n)$ is generated by the metric and the determinant.
I am interested in the case where we have a metric of indefinite signature. In particular, I would like to know whether the above theorem holds with $O(n)$ and $SO(n)$ replaced with $O(n,m)$ and $SO(n,m)$.
I have only been able to find proofs of the positive-definite case, and it seems to be folk-lore that the indefinite case holds as well, but I would like to see a proof or a reference to a proof. Perhaps its a simple corollary of the positive definite case, but I am not knowledgeable enough to see it. Any help would be appreciated.