  The pseudo-unitary group ${\bf U}\left(p,q\right)$ of signature
$\left(p,q\right)$ is the group of matrices that preserve the indefinite
pseudo-Euclidean metric on the vector space $\mathbb{C}^{p,q}$. The goal of
this paper is to describe the set ${\bf U}_{s}\left(p,p\right)$ of Hermitian,
or, self-adjoint elements in ${\bf U}\left(p,p\right)$.
