Classification of Exceptional Lie Algebras E6, E7, E8 via Dynkin Diagrams and Octonions — E8 Intelligence Research

FINDING: Classification of root systems for exceptional Lie algebras E6, E7, E8 via Dynkin diagrams and explicit bracket construction using triality and octonions. | MATH: Root system classification yields Dynkin diagrams: E6 (nodes: 6, Coxeter number 12), E7 (nodes: 7, Coxeter number 18), E8 (nodes: 8, Coxeter number 30). The bracket for E8 is given by explicit formula based on Barton-Sudbery description using oct-octonions (tensor product of octonions). Subalgebras E6 and E7 are derived from this. Key constants: dimensions — E6 (78), E7 (133), E8 (248). Root system symmetries involve angles of 60°, 90°, 120°, 150° (π/3, π/2, 2π/3, 5π/6) with root length ratios 1:√2:√3 in E8. | CONNECTION: Root systems are crystallographic — all angles and length ratios correspond to lattice symmetries. E8 root system is intimately linked to the 8-dimensional E8 lattice, the most symmetric sphere packing in 8D (kissing number 240). The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the Coxeter plane pro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com