Motivation for Hua's identity

Hua's identity is used to prove that any additive map of a division ring into itself preserving inverses must be an automorphism or antiautomorphism. His identity puts the Jordan triple product $aba$ in terms of additions and inverses, hence showing that those maps are also Jordan automorphisms; but Jordan isomorphisms between simple algebras had been shown by Ancochea and Kaplansky (1947) to be either automorphisms or antiautomorphisms, and the result for general division rings was proved by Hua himself (1949), completing the proof for additive maps preserving inverses. The study of contexts in which Jordan homomorphisms $f:R\rightarrow S$ for $R,S$ associative rings can be described in terms of (associative) homomorphisms and antihomomorphisms has a long and rich history (e.g., Jacobson and Rickart for $R$ a matrix ring, Bresar for $S$ a semiprime ring).

1. Le théorème de von Staudt en géométrie projective quaternionienne (1942). Ancochea.

2. On semi-automorphisms of division algebras (1947). Ancochea.

3. Semi-automorphisms of rings (1947). Kaplansky.

4. On the automorphisms of a sfield (1949). Hua.

5. Jordan homomorphisms of rings (1950). Jacobson, Rickart.

6. Jordan mappings of semiprime rings I, II (1989-91). Bresar.

Either this answer to my own question encapsulates the reason that Hua's identity is a big deal, or is a trivial consequence of the true raison d'être for the identity. I do not know which...

The video proves a version of the identity formatted so that the product $aba$ is expressed exclusively in terms of inverses and sums: $$aba = \left[ (a-b^{-1})^{-1}-a^{-1} \right]^{-1}+a$$ Is that it?