Let be an orthonormal basis. Let denote the group of cyclic permutations of . Then

is the standard representation of a Root System of type

with basis

The set of simple roots orthogonal to the Highest Root is and the corresponding Parabolic set is . Its complement is a 5-dimensional Heisenberg Set

The 4 maximal Abelian subsets are permuted by the canonical involution of the Heisenberg Radical hence we may choose as representatives those which contain :

The Abelian relations for these sets are generated by

The corresponding polynomial relations in variables are