$S_a$ is obviously self-adjoint and the characteristic polynomial of $$ S_a=\frac12\left(\begin{array}{cc} a_3&a_1-ia_2\\ a_1+ia_2&-a_3 \end{array}\right) $$ is given by $\tfrac14(4z^2-|a_1+ia_2|^2-a_3^2)=z^2-1/4$ and thus its eigen-values are $\pm1/2$. Hence the eigen-values of $U(t)\colon=\exp(-itS_a)$ are $e^{\pm it/2}$, i.e. $U(t)\in\UU(2)$ and $\det U(t)=1$.