John Carlos Baez on Nostr: - if you take the Cayley-Dickson construction of the octonions here: you get the ...

- if you take the Cayley-Dickson construction of the octonions here:

https://en.wikipedia.org/wiki/Split-octonion#Cayley%E2%80%93Dickson_construction

you get the ordinary octonions if ℓ²=−1 and the split octonions if ℓ² = +1. So, if you complexify them you get the same thing. For example, if you start with the complexified octonions, with ℓ²=−1, you can define ℓ' = iℓ, and the square of this will be +1.