Communitive Octonion Multiplication

[Catullus]

What would happen if you defined a communitive multiplication for octonions as k^(log(k,a)+log(k,b)), for some k such as sqrt(2)?

[JmsNxn]

What would happen if you defined a communitive multiplication for octonions as log(k,k^a+k^b), for some k such as sqrt(2)?

Something ridiculous. You would still have a noncommutative ring of octonions. But now you attach a secondary commutative group, written based off of analytic operators you'd have to prove exist in the first place. I imagine asking this for quaternions would be hard enough. Instead of asking, perhaps look into how to solve it of your own accord.