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Twisted Bhargava Cubes

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.