  Let G be a reductive group and P=MN a maximal parabolic subgroup. The group M
acts, by conjugation, on N/[N,N]. It is well known that, over an algebraically
closed field, the group M acts transitively on a Zariski open set. However,
over a general field, the structure of orbits may be quite non-trivial. A
description may involve unexpected invariants. A notable example is when G is a
split, simply connected group of type D_4, and P is the maximal parabolic
corresponding to the branching point of the Dynkin diagram. The space N/[N,N]
is also known as the Bhargava cube, and it was the starting point of his
investigations of prehomogeneous spaces. We consider a version of this problem
for the triality D_4.
