Motivation or intuition the identity $a=(b-(a-1)^{-1}b(a-1))(a^{-1}ba-(a-1)^{-1}b(a-1))^{-1}$, when $ab\neq ba$

Question

In proving the Cartan-Brauer-Hua theorem, Hua uses an obscure identity. That is: $$ a=(b-(a-1)^{-1}b(a-1))(a^{-1}ba-(a-1)^{-1}b(a-1))^{-1} $$ for $a$ and $b$ such that $ab \neq ba$. All these elements belong to a division ring. This identity confused me.I want to know some motivation or intuition about this identity. Thanks for any help.