  A transformation is obtained which completes the unification of
quadrirational Yang-Baxter maps and known integrable multi-quadratic quad
equations. By combining theory from these two classes of quad-graph models we
find an extension of the known integrability feature, and show how this leads
subsequently to a natural extension of the associated lattice geometry. The
extended lattice is encoded in a birational representation of a particular
sequence of Coxeter groups. In this setting the usual quad-graph is associated
with a subgroup of type BC_n, and is part of a larger and more symmetric
ambient space. The model also defines, for instance, integrable dynamics on a
triangle-graph associated with a subgroup of type A_n, as well as finite
degree-of-freedom dynamics, in the simplest cases associated with affine-E6 and
affine-E8 subgroups. Underlying this structure is a class of biquadratic
polynomials, that we call idempotent, which express the trisection of elliptic
function periods algebraically via the addition law.
