Suppose that $(V,b)$ is a real even-dimensional ($n=2k$) vector space with a nondenegenerate symmetric bilinear form $b$.
Question : Is there (always) a linear map $J:V\to V$ such that
- $J^2 = -I$
- $J\in O(V,b)$
I know that in the case of a positive-definite form $b$, such a (orthogonal complex) structure always exists, but I was not able to see how to extend that (or prove that we cannot extend that) to the non-positive-definite case, or find references for that question.