Phenomenological QED model in the minimally complete Geometric Representation of Clifford Algebra - What is the Gauge Group?

In geometric representation of Clifford algebra, Pauli matrices are the basis vectors of 3D space, Dirac those of 4D spacetime. Vacuum wavefunction is comprised of the Pauli algebra's 8 components - 1 scalar point, 3 vector lines (orientational degrees-of-freedom), 3 bivector areas, and 1 trivector volume. Interaction is modeled by the Clifford product, sum of dimension-reducing dot and increasing wedge products. The 8 wavefunction components one might take to be E8 root vectors, gauge group generators via the Clifford product.

The 4 fundamental constants (electric charge quantum e, Planck's angular momentum quantum h, speed of light c, and magnetic permeability of space μ) that define the QED coupling constant α permit assigning unique geometrically and topologically appropriate electric and magnetic flux quanta to each of the 8 vacuum wavefunction components. This removes the 3-fold degeneracy of vector and bivector, and increases dimensionality to 10D. Time emerges from interactions, via wedge products' dimension-increasing property. Taking quantum interaction impedances of the modes to be components of the 8 x 8 x 8 S-matrix makes it the gauge group as well. Do we call this E8 x E8 x E8?

In physical spacetime the S-matrix has 6 degrees-of-freedom, 3 each space and phase. Each orientation requires its own phase. Time is the integral of phase, collapses 6D phase space to 4D spacetime. Do we call the physical S-matrix E6, the largest possible unification group according to Georgi? Is Weinberg's preference for E8 x E8 compacted to E6 realized in this approach?