Real double flag variety for the symmetric pair $(U(p,p),G{L}_p(\mathbb{C}))$ and Galois cohomology

Let $G$ be the indefinite unitary group $U(p,p)$, $H\simeq G{L}_p(\mathbb{C})$ its symmetric subgroup, $P_S$ the Siegel parabolic subgroup of $G$, and $B_H$ a Borel subgroup of $H$. In this article, we give a classification of the orbit decomposition $H\mathrm{\backslash }(H\mathrm{/}{B}_H\times G\mathrm{/}{P}_S)$ of the real double flag variety by using the Galois cohomology in the case where $p=2$.