**Question:** What are the top dimensional cones of the 2nd Voronoi decomposition of the space of positive definite forms in $4$ variables?

The 2nd Voronoi decomposition of the cone of positive definite in $g$ variables is a $\operatorname{GL}_{g}(\mathbb{Z})$-invariant decomposition into convex cones. When $g=4$, there are suppose to be 3 top-dimensional cones up to $\operatorname{GL}_{4}$-equivalence. One cone is the principal cone spanned by the forms $$\{ x_i^2, (x_i-x_j)^{2} \}$$ (or rather the intersection of their span with the cone of positive definite forms). The other two cones can be taken to be cones subdividing the cone spanned by $$\{ x_i^2, (x_1-x_3)^2, (x_1-x_4)^2, (x_2-x_3)^2, (x_2-x_4)^2, (x_3-x_4)^2, (x_1+x_2-x_3)^2, (x_1+x_2-x_4)^2, (x_1+x_2-x_3-x_4)^{2} \}$$ (or rather the intersection of their span with the cone of positive definite forms).

But which two cones? What is the subdivision?

**P.S.** Recall the definition of the 2nd Voronoi decomposition. Suppose that we are given a positive definite quadratic form $q$ in $g$ variables. If $v \in \mathbb{R}^{g}$ is a vector, then let $w_1, \dots, w_n \in \mathbb{Z}^{g}$ be the lattice points nearest to $v$ with respect to the metric defined by $q$ and define their convex hull to be $D(a)$. The $D(a)$'s form a polyhedreal decomposition of $\mathbb{R}^{g}$ called the Delony decomposition.

We say that two forms $q_1$ and $q_2$ are Voronoi-equivalent if their define the same Delony decomposition of $\mathbb{R}^{g}$. The closures of the resulting equivalence classes are the cones defining the 2nd Voronoi decomposition of the cone of positive definite forms.

**P.P.S.:** I am tagging this as algebraic geometry because this fan comes up in the study of moduli of abelian varieties.