  I investigate the general extension of Einstein's gravity by considering the
third rank non-metricity tensor and the torsion tensor. The minimal coupling to
Dirac fields faces an ambiguity coming from a severe arbitrariness of the
Fock-Ivanenko coefficients. This arbitrariness is fed in part by the covariant
derivative of Dirac matrices, which is not completely determined as well. It is
remarkable that this feature is not exclusive to the non-metricity case: it
happens also for gravity with torsion alone. Nevertheless, theory in vacuum is
well defined and non-trivial, where torsion is the source of non-metricity or
vice-versa. I point also to the existence of two independent non-metricities.
