Hua's identity

In algebra, Hua's identity^[1] named after Hua Luogeng, states that for any elements a, b in a division ring, $${\displaystyle a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba}$$ whenever ${\displaystyle ab\neq 0,1}$. Replacing ${\displaystyle b}$ with ${\displaystyle -b^{-1}}$ gives another equivalent form of the identity: $${\displaystyle \left(a+ab^{-1}a\right)^{-1}+(a+b)^{-1}=a^{-1}.}$$