  The concept of pure spinor 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 twistor equation to the integrability of maximally isotropic
distributions.
