I will summarize nearly 40 years of work whose goal has been to use the octonions to describe the fundamental particles of nature. This talk will emphasize the underlying construction of e8, the largest of the exceptional Lie algebras, as 3x3 anti-Hermitian matrices over (the tensor product of two copies of) the octonions.

No prior knowledge of Lie algebras (or particle physics) will be assumed.

This talk is based on joint work with Corinne Manogue and Robert Wilson which has been published in a recent series of papers:
JMP 63, 081703 (2022); arXiv:2204.05310
IIG 20, 611-634 (2023); arXiv:2204.04996
JMP 65, 031702 (2024); arXiv:2309.00078
JMP 65, 031703 (2024); arXiv:2401.10534