The Local Equivalence Problem for 7-dimensional, 2-nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type

We apply E. Cartan’s method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds M up to local CR equivalence in the case that the cubic form of M satisfies a certain symmetry property with respect to the Levi form of M. The solution to the equivalence problem is given by a parallelism on a principal bundle over M which takes values in su(2, 2) or su(3, 1), depending on the signature of the nondegenerate part of the Levi form. Differentiating this parallelism provides a complete set of local invariants of M. We exhibit an explicit example of a real hypersurface in C^4 whose invariants are nontrivial