papalex on Nostr: Ah, I just found your discussion with John Carlos Baez , now I understand your ...

Ah, I just found your discussion with John Carlos Baez (npub1nf4…nqe4) , now I understand your questions 😅

I will check whether these guys are alternative tomorrow, here its way too late already. But reading your discussion it makes sense to me now that the alternative law seems to hold: If you don't do conjugations then the associator term seems to vanish (for the same reasons as in the real case). Which is not surprising as we only work over a different field now. But when we also introduce the conjugation, then we changed more than just the field! Because working over ℂ instead of ℝ would mean that we only conjugate the octonion part, but not the complex part. Because the complex conjugation is an operation on a real algebra (because it defines the real axis). Thus, terms involving the conjugation may differ from standard octonion reasoning while terms without conjugation (like the alternative laws) should carry over from octonions.

I hope for now this helps for some intuition