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 $$ab \ne 0, 1$$. Replacing $$b$$ with $$-b^{-1}$$ gives another equivalent form of the identity: $$\left(a + ab^{-1}a\right)^{-1} + (a + b)^{-1} = a^{-1}.$$

Hua's theorem
The identity is used in a proof of Hua's theorem, which states that if $$\sigma$$ is a function between division rings satisfying $$\sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^{-1}) = \sigma(a)^{-1},$$ then $$\sigma$$ is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.

Proof of the identity
One has $$(a - aba)\left(a^{-1} + \left(b^{-1} - a\right)^{-1}\right) = 1 - ab + ab\left(b^{-1} - a\right)\left(b^{-1} - a\right)^{-1} = 1.$$

The proof is valid in any ring as long as $$a, b, ab - 1$$ are units.