Clifford Algebra Unifies Quaternionic Root Systems of Coxeter Groups — E8 Intelligence Research

FINDING: Clifford algebra Cl(3,0) provides a unified geometric framework for quaternionic root systems of Coxeter groups, including B3, revealing reflection symmetries as geometric products rather than abstract matrices. | MATH: Cl(3,0) basis {1, e₁, e₂, e₃, e₁e₂, e₂e₃, e₃e₁, e₁e₂e₃}; quaternionic root systems for rank-3/4 Coxeter groups (A₃, B₃, H₃, F₄, H₄) and E₈; reflection via sandwich product r → −v r v⁻¹; B3 root system: 18 roots, Weyl group order 48, longest root squared = 2; quaternionic representation: roots as pure unit quaternions {±i, ±j, ±k, (±1±i±j±k)/2, (±i±j)/√2} | CONNECTION: B3 root system is the symmetry of the cube/octahedron — crystallographic, with Coxeter number 6; its quaternionic form embeds in Cl(3,0) where the pseudoscalar e₁e₂e₃ squares to −1, linking to H₃ (icosahedral, non-crystallographic) via the golden ratio φ = (1+√5)/2 ≈ 1.618; the paper explicitly shows B3 and H₃ share the same Clifford algebra structure, with H₃ roots containing φ and 1/φ = 0.618; t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com