Hua's identity

In algebra, Hua's identity named after Hua Luogeng, states that for any elements a, b in a division ring,

a-\left(a^{-1}+\left(b^{-1}-a\right)^{-1}\right)^{-1}=aba

whenever $ab\neq 0,1$ . Replacing $b$ with $-b^{-1}$ gives another equivalent form of the identity:

\left(a+ab^{-1}a\right)^{-1}+(a+b)^{-1}=a^{-1}.

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