Whose work this isCredit

This site exists because of one paper, and that paper exists because of thirty-four others. All of them are below, with links.

The paper this reads

A. Garrett Lisi, Division Algebras, Triality, and Exceptional Magic, arXiv:2609.12112v1 [math-ph], 10 September 2026. HTML · PDF · CC BY-NC-ND 4.0.

That licence allows no derivatives, so nothing here is a copy of it. The mathematics is nobody's property: every table, picture and number on this site is computed in this repository from the definitions, and the paper is named wherever it is the reason a thing is here. Read it; it is better than this page.

What was done here, and what was not

The mathematics is not mine and not the paper's — it runs from Hamilton and Graves in the 1840s through Cartan, Freudenthal, Tits and Vinberg.
The paper is the reason this site picks these topics, in this order, with these decompositions. Where a page follows its §, the page says so.
Every table, figure, number and check on this site was computed in this repository from the definitions. The multiplication tables in tools/algebra.py are transcribed from the paper's §2, which prints the standard ones, and are then verified against the laws they have to satisfy.
No text, equation image or figure from the paper is reproduced here. Its licence is CC BY-NC-ND 4.0, which permits no derivatives, and this site is not one.
Where physics is proposal rather than result, the proposer is named and the objections are on the same page.

The paper's references

Lisi's own list, transcribed with its links. The right-hand column is this site's note on what each one is here for.

who	year	what	why it is on the list
J. Baez	2002	The Octonions Bull. Amer. Math. Soc. 39	The standing introduction to the octonions and the magic square. Lisi's paper says it follows this one, in more painful detail.
C. H. Barton and A. Sudbery	2003	Magic squares and matrix models of Lie algebras Adv. Math. 180, 596–647	Where the magic square gets its matrix models.
L. Boyle and S. Farnsworth	2020	The standard model, the Pati-Salam model, and 'Jordan geometry' New J. Phys. 22	Another route from the exceptional algebras into particle physics.
D. Chester, A. Marrani, D. Corradetti, R. Aschheim and K. Irwin	2023	Dixon-Rosenfeld lines and the standard model Eur. Phys. J. C 83 (849)	Division algebra lines and the standard model.
D. Chester, M. Rios and A. Marrani	2023	Beyond the standard model with six-dimensional spinors Particles 6 (1)	Six-dimensional spinors past the standard model.
J. Distler and R. Garibaldi	2010	There is no 'theory of everything' inside E8 Comm. Math. Phys. 298 (2), 419–436	The standing objection to fitting three generations in E8. The site's physics page states it; Lisi §10 and §12 answer it.
G. M. Dixon	1994	Division algebras: octonions, quaternions, complex numbers, and the algebraic design of physics Kluwer	The book that made C ⊗ H ⊗ O a programme.
A. Douglas and J. Repka	2014	The gravigut algebra is not a subalgebra of E8, but E8 does contain an extended gravigut algebra SIGMA 10 (072)	What does and does not embed.
T. Dray and C. A. Manogue	2010	Octonions, E6, and particle physics J. Phys. Conf. Ser. 254 (012005)	E6 as the minimal exceptional home.
T. Dray and C. A. Manogue	2015	The geometry of the octonions World Scientific	Where su(3, D) as a way of writing these algebras comes from.
M. Dubois-Violette and I. Todorov	2018	Exceptional quantum geometry and particle physics II Nucl. Phys. B 938	The Jordan-algebra line of attack.
J. Evans	2009	Trialities and exceptional Lie algebras: deconstructing the magic square	Triality as the thing the magic square is made of — the paper's §4 and §5 lean on it, and so does this site's square.
H. Freudenthal	1964	Lie groups in the foundations of geometry Adv. Math. 1, 145–190	One of the two people the magic square is named for.
N. Furey and M. J. Hughes	2022	Division algebraic symmetry breaking Phys. Lett. B 831	Symmetry breaking done with division algebras.
N. Furey and M. J. Hughes	2025	Three generations and a trio of trialities Phys. Lett. B 865	Three generations from three trialities — the idea this site's triality page is circling.
A. B. Gillard and N. G. Gresnigt	2019	Three fermion generations with two unbroken gauge symmetries from the complex sedenions Eur. Phys. J. C 79 (446)	Someone did go past eight, on purpose.
A. Kollross	2020	Octonions, triality, the exceptional Lie algebra F4, and polar actions on the Cayley hyperbolic plane Int. J. Math. 31 (07)	F4 and triality.
K. Krasnov	2022	Spin(11,3), particles, and octonions J. Math. Phys. 63 (031701)	A different signature, same octonions.
A. G. Lisi	2007	An exceptionally simple theory of everything	The 2007 E8 paper. The particle assignment on this site's physics page is the one it made.
A. G. Lisi	2010	An explicit embedding of gravity and the standard model in E8 Representation Theory and Mathematical Physics, Contemp. Math. 557	The embedding written out.
A. G. Lisi	2015	Lie group cosmology	The setting the 2026 paper puts its superconnection in.
A. G. Lisi	2024	C, P, T, and triality	The quaternion group inside the CPT group, and its extension by triality.
P. Lounesto	2001	Clifford algebras and spinors Cambridge, 2nd ed.	Reflections in a Clifford algebra, the standard account.
C. A. Manogue, T. Dray and R. A. Wilson	2022	Octions: an E8 description of the standard model J. Math. Phys. 63 (081703)	Another E8 description.
C. C. Perelman	2021	On Jordan-Clifford algebras, three fermion generations with Higgs fields and a SU(3)×SU(2)L×SU(2)R×U(1) model Adv. Appl. Clifford Algebr. 31 (53)	Three generations from a Jordan-Clifford algebra.
I. R. Porteous	1995	Clifford algebras and the classical groups Cambridge	The other standard reference for reflections.
P. Ramond	2003	Exceptional groups and physics Groupe 24 plenary talk	The overview talk.
J. Tits	1966	Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles Nederl. Akad. Wetensch. Proc. Ser. A 69, 223–237	The other person the magic square is named for.
V. Vaibhav and T. P. Singh	2023	Left-right symmetric fermions and sterile neutrinos from complex split biquaternions and bioctonions Adv. Appl. Clifford Algebr. 33 (32)	Split algebras, put to work.
E. B. Vinberg	1976	The Weyl group of a graded Lie algebra Math. USSR-Izv. 10 (3)	Where the Θ-groups on this site's E8 page come from. Russian original: Izv. Akad. Nauk SSSR Ser. Mat. 40 (3) (1976).
R. A. Wilson, T. Dray and C. A. Manogue	2023	An octonionic construction of E8 and the Lie algebra magic square Innov. Incidence Geom. 20	Building E8 out of octonions.
R. A. Wilson	2024	On possible embeddings of the standard model of particle physics and gravity in E8	What can be embedded, carefully.
P. Woit	2021	Euclidean twistor unification	Twistors meet the same incidence relation triality produces.
J. A. Wolf and A. Gray	1968	Homogeneous spaces defined by Lie group automorphisms. I J. Diff. Geom. 2, 77–114	3-symmetric spaces — what a triality automorphism leaves behind.

This site

Text and figures: Nan · hongdam.net · CC BY 4.0 — CC BY 4.0. Credit line: Nan · hongdam.net · CC BY 4.0, with a link to https://nanobotco.github.io/exceptional-magic/.
Code: MIT — tools/, js/, tests/.
Source: github.com/NaNoBotCo/exceptional-magic
Built: 2026-09-23, from 139 passing checks

Reading order, if you want the mathematics properly

Baez, The Octonions (2002) — the place to start, and the paper Lisi's follows. J. Baez 2002
Dray and Manogue, The Geometry of the Octonions (2015) — book length, worked out.
Evans, Trialities and exceptional Lie algebras (2009) — the magic square taken apart. J. Evans 2009
Lisi, Division Algebras, Triality, and Exceptional Magic (2026) — the one this site reads, and the most explicit of them about how to actually compute with triality.