papalex on Nostr: The equation holds and is in fact super straight forward (one might say trivial) to ...

The equation holds and is in fact super straight forward (one might say trivial) to prove from alternative algebra.

To avoid confusion, the amendment equation reads:

(𝑥𝑥*)𝑦−𝑥(𝑥*𝑦)=(𝑦𝑥)𝑥*−𝑦(𝑥𝑥*)

However, this follows immediately from the fact that in any alternative algebra 𝔸 and 𝑎,𝑏,𝑐∈𝔸 it holds:

(𝑎𝑏)𝑐−𝑎(𝑏𝑐)=(𝑐𝑎)𝑏−𝑐(𝑎𝑏)

or using the associator:

[𝑎,𝑏,𝑐]=[𝑐,𝑎,𝑏]

since

[𝑎,𝑏,𝑐]=−[𝑎,𝑐,𝑏]=[𝑐,𝑎,𝑏]

See for example https://en.wikipedia.org/wiki/Alternative_algebra#The_associator

Thus, since the octonions and the bioctonions are alternative, your amendment is correct by setting 𝑎=𝑥,𝑏=𝑥*,𝑐=𝑦.

PS: John Carlos Baez (npub1nf4…nqe4) either you edited your post, or more likely my phone did not render the maths correctly. Now on my computer it look the same 👍