[gowld]

Are there 32-onions that lack power associativity?

https://en.wikipedia.org/wiki/Power_associativity

https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...

[bonzini]

No, power associativity is never lost, after sedenions the properties remain the same.

[OscarCunningham]

They don't remain exactly the same. The answers to this (https://math.stackexchange.com/questions/641809/what-specifi...) question provide some interesting starting-points.