  We study the anisotropy theorem for Stanley-Reisner rings of simplicial
homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem
implies the Hard Lefschetz theorem as well as McMullen's g-conjecture for such
spheres. Our first result is an explicit description of the quadratic form. We
use this description to prove a conjecture stated by Papadakis and Petrotou.
All anisotropy theorems for homology spheres and pseudo-manifolds in
characteristic 2 follow from this conjecture. Using a specialization argument,
we prove anisotropy for certain homology spheres over the field $\mathbb{Q}$.
These results provide another self-contained proof of the g-conjecture for
homology spheres in characteristic 2.
