  We study the magnetic order and excitations in strong spin-orbit coupled, Van
Vleck-type, $d^4$ Mott insulators on a square lattice. Extending the previous
work, we include the tetragonal crystal field splitting and explore its effects
on magnetic phase diagram and magnon spectra. Two different ordered phases,
with in-plane and out-of-plane orientation of the staggered moments, are found
for the higher and lower values of the crystal field splitting, respectively.
The magnetic excitation spectra for paramagnetic and magnetically ordered
phases are calculated and discussed in the context of a candidate spin-orbit
$d^4$ Mott insulator Ca$_2$RuO$_4$.
