Octo-Octonion Triality Realizes E8 via Explicit Freudenthal–Tits Brackets — E8 Intelligence Research

FINDING: The Freudenthal–Tits magic square construction of exceptional Lie algebras E6/E7/E8 is explicitly realized via triality and octo‑octonions, with an explicit bracket formula for E8 and its E6/E7 subalgebras. | MATH: The magic square is a 3×3 (or 4×4) array of Lie algebras built from division algebras (R, C, H, O) and their Jordan algebras. For E8: the Barton–Sudbery decomposition gives \(e_8 = so(8) ⊕ so(8) ⊕ 8_v ⊗ 8_s ⊕ 8_v ⊗ 8_c ⊕ 8_s ⊗ 8_c\) (triality triple of 8‑dimensional representations). The explicit bracket uses the octonionic structure constants \(cᵢⱼₖ\) (totally antisymmetric, satisfying \(cᵢⱼₘcₖₗₘ = δᵢₖδⱼₗ - δᵢₗδⱼₖ + εᵢⱼₖₗ\) with \(ε\) the 4‑form). E7 arises as \(e_7 = so(12) ⊕ su(2) ⊕ 32\) (spinor), E6 as \(e_6 = so(10) ⊕ u(1) ⊕ 16 ⊕ 16̄\) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com