John Carlos Baez on Nostr: Actually this is a request for help! You can help if you're good at calculating with ...

Actually this is a request for help! You can help if you're good at calculating with 'bioctonions'. These are elements of the algebra ℂ⊗𝕆, the tensor product of the complex numbers and octonions. If you know about tensor products of (nonassociative) algebras, this is all you need to know.

Bioctonions are a nonassociative *-algebra, where we define

(a ⊗ b)* = a* ⊗ b* for all a ∈ ℂ, b ∈ 𝕆.

My question is this. Given bioctonions x and y, do we always have

(xx*) y = x (x* y) ?

This is true for octonions even though they are not associative. I suspect it's not true for bioctonions. But it would make me incredibly happy if it were true. Latham Boyle's theory of particle physics would be improved by this fact... if it were a fact!