Formal ring of a cubic solid angle

A rational cubic form with real plane factors determines a solid angle for whose lattice points a formal ring can be constructed and can be made invariant under units. As in the first half of this work (in Volume 8 of this Journal), there is a finite basis for the invariant ring but peculiar effects occur in higher dimension. There might be, for instance, no minimal basis subdivision for the support polyhedron, and no cyclic basis, (simulating algebraic resolution difficulties in C⁸). Furthermore, the support polyhedron is “flat” in some affine sense. Cases are computed for small discriminant cubics.