papalex on Nostr: Happy to help! I was also just writing down the TLDR version for the intuition but ...

Happy to help! I was also just writing down the TLDR version for the intuition but didn't have the place left in the post, so here we go:

Two things are crucial:
1) First \(y\) lies outside the quaternionic subalgebra spanned by \(x\) to obtain a non-trivial associator,
2) and second, the complex factor to each term in \(x\) counters -- due to complex conjugation -- the anti-symmetry of the associator.