  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 ${\rm LG}(n,2n)$ and over the complex
numbers the image is defined, 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 ${\rm
Spin}(2n+1)$. 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.
