Torsion parallel pure spinors on neutral manifolds

We study irreducible real pure spinors on pseudo-Riemannian manifolds of neutral signature using the theory of real spinorial forms. We prove that the square of such a spinor is a decomposable differential form of middle degree satisfying a natural duality condition. In signature $(4,4)$, we show that non-pure spinors correspond to ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_0(4,3)$-structures, yielding an intrinsic algebraic characterization of these structures. In addition, we characterize real pure spinors parallel with respect to metric connections with torsion in terms of an equivalent differential system for their squares. As an application, we study left-invariant supersymmetric solutions of the NS-NS supergravity system on certain four-dimensional Lie groups.