Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some particular class of matrices, which is considerably smaller than the space $O(n,\mathbb C)$. (So a bit similar to, for example, the fact that any symmetric matrix $Q$ can be written as $T^{-1}ST$, where $T\in O(n,\mathbb R)$ and $S$ is a diagonal matrix.)
More generally, I would be helped to know the answer to the same question for the case that $T$ belongs to the indefinite orthogonal group $O(p,q,\mathbb R)\subset O(p+q,\mathbb C)$. 
Thanks for any help or references.
 A: Some thoughts on your issue. 
It is known that complex matrices $A$ and $B$ are orthogonally similar if and only if the pairs $(A,A^t)$ and $(B,B^t)$ are simultaneously similar through an invertible matrix with real entries (one implication is straightforward, for the converse consider an invertible real matrix $P$ such that $A=PBP^{-1}$ and $A^t=PB^t P^{-1}$, use the polar decomposition $P=OS$ and show that $S$ commutes with $B$). 
Thus, since you are dealing with orthogonal complex matrices, your problem amounts to classifying complex orthogonal matrices up to conjugation with a real-valued invertible matrix. Likewise, complex matrices $A$ and $B$ are real-similar if and only if the pairs
$(Re(A),Im(A))$ and $(Re(B),Im(B))$ of real matrices are simultaneously similar. 
Thus, one would need some kind of a canonical form of pairs $(M,N)$ of real matrices such that $M+iN \in O_n(\mathbb{C})$, under simultaneous similarity. This should not be easy, as suggested by the literature on the simultaneous similarity problem 
