Pure subspaces, generalizing the concept of pure spinors

The concept of pure spinors is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the choice of a spinorial connection is exploited in order to relate the twistor equation to the integrability of maximally isotropic distributions.