Exceptional Lie Algebras Classified by Root Systems and E8's Octonionic Triality — E8 Intelligence Research

FINDING: The exceptional Lie algebras G2, F4, E6, E7, E8 are classified by their crystallographic root systems, with E8's bracket explicitly constructed via triality and octonions, linking root-length ratios to the golden ratio. | MATH: Root system length ratios: G2: long/short = √3 (≈1.732); F4: √2 (≈1.414); E6, E7, E8: all roots equal length (ratio 1). Casimir operator eigenvalues for E8 (adjoint, dim 248): C₂ = 60 (in units where long root² = 2). E8 root system: 240 roots, rank 8, Weyl group order 696,729,600. Explicit bracket formula (arXiv:2504.16513v3) uses triality: E8 ≅ so(8) ⊕ 8ᵥ ⊕ 8ₛ ⊕ 8ₜ (Barton–Sudbery), with octonionic structure constants. | CONNECTION: E8's root system contains the golden ratio via its Coxeter number h=30: 2cos(π/h) = 2cos(6°) ≈ 1.989 — not 1.618. However, the ratio of the squared lengths of the two root lengths in G2 is 3, and in F4 is 2 — both crystallographic. The golden ratio appears in E8's Weyl orbit structure: the ratio of the number of roots at an Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com