Differential spinors for $\mathrm{G}_2^*$ and isotropic structures

We obtain a correspondence between irreducible real differential spinors on pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions to an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the K\"ahler-Atiyah bundle of $(M,g)$. In particular, we obtain an intrinsic algebraic characterization of $\mathrm{G}_2^*$-structures and we provide the first explicit characterization of isotropic irreducible spinors in signature $(4,3)$ parallel under a general connection on the spinor bundle, which we apply to the spinorial lift of metric connections with torsion.