Okubo'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.
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. Assuming A has a multiplicative unity, define involution and right and left multiplication operators by Evidently is an involution and preserves the quadratic form.
The overline notation stresses the fact that complex and quaternion conjugation are partial cases of it.