papalex on Nostr: I ask myself the same question. Though did not yet have the time to think about it. ...

I ask myself the same question. Though did not yet have the time to think about it. My guess for a simpler counter example would be a combination of x and y consisting of just two octonions each which form different quaternionic subalgebras within the octonions (usually all "exceptional" examples are of this type I feel). I started with three octonions mainly because I followed my gut feeling rather than systematic search and because I wasn't sure whether 1 and i as complex coefficients would be sufficient. But it is certainly simpler to check such an example, its just 12 terms compared to the 27 terms when multiplying out the three three-fold combinations :D