  A simplicial sphere $\Delta$ is said to be generically anisotropic over a
field $\mathbb{F}$ if, for a certain purely transcendental field extension
$\mathbf{k}$ of $\mathbb{F}$, a certain Artinian reduction $A$ of the face ring
$\mathbf{k}[\Delta]$ has the following property: For every nonzero homogeneous
element $\alpha\in A$ of degree at most $(\dim\Delta+1)/2$, its square
$\alpha^2$ is also nonzero. The importance of this property is that the hard
Lefschetz property for simplicial spheres can be derived from it. A recent
result of Papadakis and Petrotou shows that every simplicial sphere is
generically anisotropic over any field of characteristic $2$. In this paper, we
give an equivalent condition of being generically anisotropic, and use it to
present a simplified proof of Papadakis-Petrotou theorem for PL-spheres. We
also prove that the simplicial spheres of dimension $2$ are generically
anisotropic over any field $\mathbb{F}$.
