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.

Unital composition algebras are called Hurwitz algebras. If the ground field K is the field of real numbers and N is positive-definite, then A is called a Euclidean Hurwitz algebra.

If K has characteristic not equal to 2, then a bilinear form (a, b) = 1/2[N(a + b) − N(a) − N(b)] is associated with the quadratic form N.

...
Wikipedia