Octo-Octonion Triality Realizes E8 via Explicit Freudenthal–Tits Brackets — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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 e8 = so(8) ⊕ so(8) ⊕ 8v ⊗ 8s ⊕ 8v ⊗ 8c ⊕ 8s ⊗ 8c (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 e7 = so(12) ⊕ su(2) ⊕ 32 (spinor), E6 as e6 = so(10) ⊕ u(1) ⊕ 16 ⊕ 16̅
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026octooctonion,
title = {{Octo-Octonion Triality Realizes E8 via Explicit Freudenthal–Tits Brackets — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23464950},
url = {https://zenodo.org/records/23464950},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Octo-Octonion Triality Realizes E8 via Explicit Freudenthal–Tits Brackets — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23464950
UR - https://zenodo.org/records/23464950
AB - 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 c_ijk (totally antisymmetric, satisfying c_ijmc_klm = δ_ikδ_jl − δ_ilδ_jk + ϵ_ijkl 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 ⊕ 16̅ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -