  We point attention to the fact that in SO(10) models with non-canonical (type
II) see-saw mechanism and exact b-tau unification the trace of the neutrino
mass matrix is very small, in fact practically zero. This has the advantage of
being a basis independent feature. Taking a vanishing trace as input, immediate
phenomenological consequences for the values of the neutrino masses, the CP
phases or the predictions for neutrinoless double beta decay arise. We analyze
the impact of the zero trace condition for the normal and inverted mass
ordering and in case of CP conservation and violation. Simple candidate mass
matrices with (close to) vanishing trace and non-zero U_{e3} are proposed. We
also compare the results with the other most simple basis independent property,
namely a vanishing determinant.
