$S$ is real symmetric. $T$ is real skew-symmetric. I have shown that $T\pm iS$ is skew-Hermitian. I am further asked to show that $U = (I+T+iS)(I-T-iS)^{-1}$ is unitary.
Denoting by $^\dagger$ the conjugate transpose, I have that
$U^\dagger = [(I+T+iS)(I-T-iS)^{-1}]^\dagger$
$= ((I-T-iS)^\dagger)^{-1}(I+T+iS)^\dagger$
$= (I+T+iS)^{-1}(I-T-iS)$
But then the products $UU^\dagger$ and $U^\dagger U$ have the factors in the wrong order to cancel out to make identity. I must be missing something obvious here; I'd appreciate a steer!