Lagrangian Grassmannians and Spinor Varieties in Characteristic Two

The vector space of symmetric matrices of size n has a natural map to a projective space of dimension 2(n) - 1 given by the principal minors. This map extends to the Lagrangian Grassmannian LG(n, 2n) and over the complex numbers the image is de fined, as a set, by quartic equations. In case the characteristic of the field is two, it was observed that, for n = 3, 4, the image is defined by quadrics. In this paper we show that this is the case for any n and that moreover the image is the spinor variety associated to Spin(2n + 1). Since some of the motivating examples are of interest in supergravity and in the black-hole/qubit correspondence, we conclude with a brief examination of other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras.