The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice $E_7$

We show that the Minkowski sum $P_{\mathrm V}(E_7)+Z(U)$ of the Voronoi polytope $P_{\mathrm V}(E_7)$ of the root lattice $E_7$ and the zonotope $Z(U)$ is a 7-dimensional parallelohedron if and only if the set $U$ consists of minimal vectors of the dual lattice $E_7^*$ up to scalar multiplication, and $U$ does not contain forbidden sets. The minimal vectors of $E_7$ are the vectors $r$ of the classical root system $\mathbf E_7$. If the $r^2$-norm of the roots is set equal to 2, then the scalar products of minimal vectors from the dual lattice only take the values $\pm1/2$. A set of minimal vectors is referred to as forbidden if it consists of six vectors, and the directions of some of these vectors can be changed so as to obtain a set of six vectors with all the pairwise scalar products equal to $1/2$.
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