Albert algebra

The set of 3×3 self-adjoint matrices over the octonions with binary operation $$ x \star y = \frac12 (x \cdot y + y \cdot x) \,, $$ where $\cdot$ denotes matrix multiplication.

The operation is commutative but not associative. It is an example of an exceptional Jordan algebra. Because most other exceptional Jordan algebras are constructed using this one, it is often referred to as "the" exceptional Jordan algebra.

References

• A. V. Mikhalev, Gunter F. Pilz (2002) "The Concise Handbook of Algebra", (Springer) ISBN 0792370724, page 346.
• Richard D. Schafer (1995) [1966], An Introduction to Nonassociative Algebras (Dover) ISBN 0-486-68813-5, page 102.