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

Abstract

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

Division Algebras, Triality, and Exceptional Magic
A. Garrett Lisi
arXiv10 Sep 2026PreprintCC BY-NC-ND 4.0Full textMathematical physics High-energy & particle physicsarXiv
Octo-Octonion Triality Realizes E8 via Explicit Freudenthal–Tits Brackets — E8 Intelligence Research
Andrew Stewart Caldin
Zenodo19 Oct 2026PreprintCC BY 4.0Zenodo
The Octonionic Rational Wave Numbers: Non-Associative Sequence Algebras, Moufang Loops, and Exceptional G₂ Symmetries
Terence R. Smith
arXiv20 Oct 2026PreprintarXiv licenceFull textAlgebra & number theory Mathematical physicsarXiv
The standard model, the exceptional Jordan algebra, and triality
Latham A. Boyle
Journal of Mathematical Physics1 Jul 2026Peer-reviewedCC BY57 citedGeneral mathematicsOpenAlex
Transferred L_∞-structures and a classification of binary brackets for finite-dimensional skew-gentle algebras
Zixu Li, Liyu Liu, Lingchao Meng
arXiv20 Sep 2026PreprintCC0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{caldin2026octonionic,
  title = {{Octonionic Triality Unifies Exceptional Lie Algebras via Magic Square E8 Brackets — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23464960},
  url = {https://zenodo.org/records/23464960},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Octonionic Triality Unifies Exceptional Lie Algebras via Magic Square E8 Brackets — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/19
PB  - Zenodo
DO  - 10.5281/zenodo.23464960
UR  - https://zenodo.org/records/23464960
AB  - 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
ER  -

Abstract

Related

Cite