 Indefinite special unitary group

The indefinite special unitary group preserves a Hermitian form of signature $(p,q)$ and has determinant one: $M^{†}JM=J$, where $J=\mathrm{diag}(I_p,-I_q)$. With both $p,q$ positive it is noncompact, unlike the group preserving a positive-definite form. Its Lie algebra satisfies $X^{†}J+JX=0$ and $\mathrm{Tr}X=0$.