On the sum of a parallelotope and a zonotope

In the case of the root lattice E6, it is possible to give a more geometric description of the admissible sets of vectors U. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of 27 lines in a cubic surface. Based on a detailed study of the geometry of PV(E6), we give a simple characterization of the configurations of vectors U such that PV(E6)+Z(U) is a parallelotope. The enumeration yields 10 maximal families of vectors, which are presented by their description as regular matroids.