The problem of understanding why neutrino mixing is different from quark mixing is of fundamental importance today. The Majorana nature of neutrinos, totally different from the Dirac nature of quarks, could easily be the source of such a difference. In the most general case of a Dirac-Majorana mass term, the Majorana nature of neutrinos implies a mixing between active and sterile states, if they exist. Here we concentrate on electron neutrinos and assume that electron neutrinos can oscillate both into active states and into sterile states, with mixing angle similar to the Cabibbo angle. This assumption, together with a squared-mass difference at the eV 2 scale, in order to account for the LSND anomaly, predicts a disappearance signal that is compatible with SNO-NC measurements in solar neutrinos and with Gallium calibration source recent results. Active-sterile oscillations of this kind could also affect electron neutrinos in MiniBoone. Comparison of this hypothesis with reactor limits suggests new measurements with future beta beams.