Pure spinors and quadric Grassmanians

If, following E. Cartan, the simplest spinors (pure) are conceived as equivalent to isotropic (or null or optical) polarized planes in complex spaces, then the most natural tensors generated (bilinearly) by the simplest spinors are isotropic vectors rather than ordinary linear ones. The conjecture that spinors are fundamental would then imply that non-linear geometry of isotropic elements should be more elementary in general than the linear one; and the relevance of optical geometry (optical flags, optical groups) on space-time manifolds for the explanation of optical phenomenology in the frame of general relativity 5] could already constitute a first confirmation of this conjecture.Only 2- and 4-component spinors build up linear spinor spaces while 8, 16, 32,...component pure spinors, instead, are subject to covariant (quadratic) constraint equations and build up non-linear sets isomorphic, up to a sign, to quadric Grassmanians and, for neutral and conformal spaces, to Lie groups.The possible relevance of such pure spinor properties for physics is conjectured and exemplified.