Paul Schwahn on Nostr: Right now I don't have an answer for the exact you posed, but are you sure this is ...

Right now I don't have an answer for the exact you posed, but are you sure this is the definition of conjugation you want?

Viewing ℂ⊗𝕆 as a composition algebra over ℂ (composition algebras satisfy a bunch of "nice" identities even if they're not associative), it has a canonically defined conjugation given by

(a ⊗ b)* = a ⊗ b*

for a ∈ ℂ, b ∈ 𝕆. With this we have xx*∈ ℂ, and the identity (xx*)y = x(x*y) holds true (see e.g. the book "Octonions, Jordan Algebras and Exceptional Groups" by Springer & Veldkamp).