Octonionic Triality Unifies Exceptional Lie Algebras via Magic Square E8 Brackets — E8 Intelligence Research

FINDING: The Freudenthal-Tits magic square construction unifies exceptional Lie algebras E6/E7/E8 via octonionic triality, with explicit E8 bracket formulas now derived from Barton-Sudbery's oct-octonion description. MATH: - Freudenthal-Tits magic square: 3×3 array of Lie algebras generated by pairs of division algebras (ℝ, ℂ, ℍ, 𝕆). Exceptional row/column: E6, E7, E8 appear at positions (𝕆,ℂ), (𝕆,ℍ), (𝕆,𝕆) respectively. - E8 bracket (arXiv:2504.16513v3): Explicit formula using triality on 𝕆⊗𝕆, with E8 = 𝔡𝔢𝔯(𝕆) ⊕ 𝔡𝔢𝔯(𝕆) ⊕ 3×(𝕆⊗𝕆) — the "triality" decomposition: 248 = 14 ⊕ 14 ⊕ 3×64 (where 64 = 8×8 octonion pairs). - E6 ⊂ E7 ⊂ E8: E6 = 78 = 26 ⊕ 26 ⊕ 26 (triality of 26-dim irreducible); E7 = 133 = 133 = (𝔰𝔩(2)⊕𝔰𝔬(8)) ⊕ 3×(2⊗8⊗8) — dimension check: 3+28+3×32 = 133. - Key constants: 248 (E8), 133 (E7), 78 (E6), 26 (Freudenthal triple system dimension), 8 (octonion dimension), 3 (triality symmetry of 𝔰𝔬(8)). CONNECTION: - The magic square's structure encodes **root system symmetries**: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com