  We consider the second-order in time Strang-splitting approximation for
vector-valued and matrix-valued Allen-Cahn equations. Both the linear
propagator and the nonlinear propagator are computed explicitly. For the
vector-valued case, we prove the maximum principle and unconditional energy
dissipation for a judiciously modified energy functional. The modified energy
functional is close to the classical energy up to $\mathcal O(\tau)$ where
$\tau$ is the splitting step. For the matrix-valued case, we prove a sharp
maximum principle in the matrix Frobenius norm. We show modified energy
dissipation under very mild splitting step constraints. We exhibit several
numerical examples to show the efficiency of the method as well as the
sharpness of the results.
