papalex on Nostr: 1) ((1+𝑖)𝑒₁)*=(−1+𝑖)𝑒₁ , in particular, there are hyperbolic ...

1) ((1+𝑖)𝑒₁)*=(−1+𝑖)𝑒₁ , in particular, there are hyperbolic elements, like (𝑖𝑒₁)*=𝑖𝑒₁. This follows immediately from the definition of the conjugation in John Carlos Baez (npub1nf4…nqe4) 's original post.

2) I haven't checked that. I would guess that not, but my longer post above seems to indicate that it might be (at least right now I don't see how to break the identity 𝑎²𝑏=𝑎(𝑎𝑏)). If you tensor the quaternions to it it certainly ceases to be alternative, but that doesn't help here.

3) Nope, by 1) you can easily construct counter examples. In fact, 𝑥𝑥* (as defined in the long post above) is not real but 2+2𝑖𝑒₄. I think though that 𝑥𝑥* always gives you a non-imaginary number, i.e. a sum of only real and hyperbolic numbers.