Neutrino mass matrices leaving no trace

We point attention to the fact that in SO(10) models with non-canonical (type II) see-saw mechanism and exact b-τ 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 Ue3 are proposed. We also compare the results with the other most simple basis independent property, namely a vanishing determinant