Separation of variables and quantum numbers for Weyl neutrino fields on curved space-time

We construct, via separation of variables, a first-order differential operator whose commutator with the Weyl neutrino operator is proportional to it in the class (mathfrak{D}) of Petrov type D vacuum and electrovac backgrounds with cosmological constant. Further, we characterize tensorially all such operators on an arbitrary background space-time. These results when combined explain the existence of a two-index Killing spinor in the class (mathfrak{D}) .