Split-octonion - Zorn's Vector-matrix Algebra

Zorn's Vector-matrix Algebra

Since the split-octonions are nonassociative they cannot be represented by ordinary matrices (matrix multiplication is always associative). Zorn found a way to represent them as "matrices" containing both scalars and vectors using a modified version of matrix multiplication. Specifically, define a vector-matrix to be a 2×2 matrix of the form

where a and b are real numbers and v and w are vectors in R3. Define multiplication of these matrices by the rule

where · and × are the ordinary dot product and cross product of 3-vectors. With addition and scalar multiplication defined as usual the set of all such matrices forms a nonassociative unital 8-dimensional algebra over the reals, called Zorn's vector-matrix algebra.

Define the "determinant" of a vector-matrix by the rule

.

This determinant is a quadratic form on the Zorn's algebra which satisfies the composition rule:

Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions. Write an octonion x in the form

where and b are real numbers and a and b are pure quaternions regarded as vectors in R3. The isomorphism from the split-octonions to the Zorn's algebra is given by

This isomorphism preserves the norm since .