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Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers. Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...

This question is trying to abstract out in a self-contained way the point that is probably being made in page 6 of this paper, https://arxiv.org/pdf/1604.03544.pdf and why Theorem 4.1 there works. ...