The Hua identity implies alternativity

The well-known Hua identity $a-(a^{-1}+(b^{-1}-a{)}^{-1}{)}^{-1}=aba$ holds in any associative division ring. We prove, in a sense, a converse: if a (nonassociative) ring with unit and multiplicative inverses satisfies the Hua identity, then it is alternative.