On seven-dimensional quaternionic contact solvable Lie groups

We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven-dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven-dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven-dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group