Okubo algebra

In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo (1978). Okubo algebras are composition algebras, flexible algebras (A(BA) = (AB)A), Lie admissible algebras, and power associative, but are not associative, not alternative algebras, and do not have an identity element.

Okubu's example was the algebra of 3 by 3 trace zero complex matrices, with the product of X and Y given by aXY + bYX – Tr(XY)I/3 where I is the identity matrix and a and b satisfy a + b = 3ab = 1. The Hermitian elements form an 8-dimensional real non-associative division algebra. A similar construction works for any cubic alternative separable algebra over a field containing a primitive cube root of unity. An Okubo algebra is an algebra constructed in this way from the trace 0 elements of a degree 3 central simple algebra over a field.
References

Hazewinkel, Michiel, ed. (2001), "Okubo_algebra", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Okubo, Susumu (1978), "Pseudo-quaternion and pseudo-octonion algebras", Hadronic J. 1 (4): 1250–1278, MR 0510100
Okubo, Susumu (1995), Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical Physics 2, Cambridge: Cambridge University Press, ISBN 9780521472159, MR 1356224

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