Real sedenions S - Knowlpedia

The real sedenions $\mathbb S$ are the real vector space $\mathbb O\oplus\mathbb O$ , with octonionic coordinates, componentwise addition, and multiplication

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

The unit is $(1,0)$ , and conjugation is $\overline{(a,b)}=(\overline a,-b)$ . This is the 16-dimensional fourth Cayley–Dickson algebra over $\mathbb R$ .

Algebraic boundaries

The product is neither associative nor alternative. Nonzero zero divisors exist, so this is not a division algebra and the positive quadratic form $N(a,b)=|a|^2+|b|^2$ is not multiplicative. The formula $x^{-1}=\overline x/N(x)$ still gives a two-sided inverse for each nonzero element; without associativity, that fact does not imply cancellation. Thus inverse formulas must not be used to register $\mathbb S$ as a field, ring, or composition algebra.

Catalogue convention

Here $\mathbb S$ always denotes sedenions, not a sphere or the split octonions. Its real vector-space and nonassociative-algebra views are distinct choices of morphisms on the same object.

References

John C. Baez, The Octonions, Section 2.2, Cayley–Dickson construction, Propositions 1–5 and the sedenion paragraph.