View source for Albert algebra

{{MSC|17C40}}

The set of 3×3 [[Self-adjoint linear transformation|self-adjoint]] matrices over the [[octonion]]s 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.