Split-octonion

In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers.

Unlike the standard octonions, they contain non-zero elements which are non-invertible.

Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers.

The octonions and the split-octonions can be obtained from the Cayley–Dickson construction by defining a multiplication on pairs of quaternions.

We introduce a new imaginary unit ℓ and write a pair of quaternions (a, b) in the form a + ℓb.

The product is defined by the rule:[1] where If λ is chosen to be −1, we get the octonions.

can be written as a linear combination of the basis elements, with real coefficients

The red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table.

If N(x) ≠ 0, then x has a (two-sided) multiplicative inverse x−1 given by The split-octonions, like the octonions, are noncommutative and nonassociative.

That is, The split-octonions satisfy the Moufang identities and so form an alternative algebra.

Therefore, by Artin's theorem, the subalgebra generated by any two elements is associative.

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.

[2] Specifically, define a vector-matrix to be a 2×2 matrix of the form[3][4][5][6] where a and b are real numbers and v and w are vectors in R3.

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 Zorn's algebra which satisfies the composition rule: Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions.

are real numbers and v and w are pure imaginary quaternions regarded as vectors in R3.