Split octonions O_s - Knowlpedia

The real split octonion algebra $\mathbb O_s$ is $\mathbb H\oplus\mathbb H$ , with quaternionic coordinates, componentwise addition, and multiplication

(a,b)(c,d)=(ac+\overline d b,\;da+b\overline c).

Its unit is $(1,0)$ , conjugation is $\overline{(a,b)}=(\overline a,-b)$ , and its multiplicative quadratic form is $N(a,b)=|a|^2-|b|^2$ .

Split form

This is an eight-dimensional alternative, nonassociative real composition algebra. Its norm has signature $(4,4)$ . The elements $(1,1)$ and $(1,-1)$ are nonzero and their product is zero, so it is not a division algebra. The plus sign in the first component distinguishes this construction from the real division octonions.

Hermitian matrices

The Hermitian degree-three construction gives the split Albert algebra. Its scalar field is real, even though some notation for its coefficient algebra resembles complex notation. “Split” records the algebra and norm form, not a change of scalar field.

References

Alberto Elduque, Composition algebras, Section 2.1, equation (7); Theorem 2.11 and Corollary 2.12.