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 \(\mathfrak{e}_8 = \mathfrak{so}(8) \oplus \mathfrak{so}(8) \oplus 8_v \otimes 8_s \oplus 8_v \otimes 8_c \oplus 8_s \otimes 8_c\) (triality triple of 8‑dimensional representations). The explicit bracket uses the octonionic structure constants \(c_{ijk}\) (totally antisymmetric, satisfying \(c_{ijm}c_{klm} = \delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk} + \epsilon_{ijkl}\) with \(\epsilon\) the 4‑form). E7 arises as \(\mathfrak{e}_7 = \mathfrak{so}(12) \oplus \mathfrak{su}(2) \oplus 32\) (spinor), E6 as \(\mathfrak{e}_6 = \mathfrak{so}(10) \oplus \mathfrak{u}(1) \oplus 16 \oplus \overline{16}\)

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com