Solution
= Solution
Using $(AB)^\dagger=B^\dagger A^\dagger$, the <conjugate transpose> of the proposed <matrix> is
$$
H^\dagger=-i(B^\dagger A^\dagger-A^\dagger B^\dagger)
=-i(BA-AB)=i(AB-BA)=H.
$$
Thus
$$
\boxed{i[A,B]\ \text{is Hermitian}.}
$$
The <commutator> of two <Hermitian matrices> is skew-Hermitian, and multiplication by $i$ changes it into a <Hermitian matrix>. No simplicity assumption on the spectra is needed for this part.