Each <Pauli matrix> is Hermitian and the coefficients $\alpha_j$ are real, so $T^\dagger=T$. Taking the <conjugate transpose> of the absolutely convergent <matrix exponential> series yields
$$
U^\dagger=e^{-iT/2}.
$$
The exponents $-iT/2$ and $iT/2$ <commute>. Therefore $U^\dagger U=e^{-iT/2+iT/2}=I$, proving that \b[$U$ is unitary].

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2015 / 12 / iii / Solution

 Solution (source code)