Hermitian parts of a unitary matrix ($A=(U+U^{†})/2,B=(U-U^{†})/(2i)$)

Every unitary matrix has a unique decomposition $U=A+iB$ into Hermitian matrices. The formulas follow by adding and subtracting $U$ and its conjugate transpose. Expanding $U^{†}U=U{U}^{†}=I$ gives $A^2+B^2+i(AB-BA)=I$ and $A^2+B^2-i(AB-BA)=I$. Therefore the parts commute and $A^2+B^2=I$.