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

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.

• Can you provide some more detail about what kind of objects $a$ and $b$ are? If everything commutes, this is not a well-defined statement. Otherwise it may be but it is still unclear which operations are permitted. – quarague Jan 28 at 10:33
• Conjugation in a group is not "obscure". – Dietrich Burde Jan 28 at 10:48
• I think whatever this identity is, it's probably equivalent to Hua's identity? If so, you're re-asking this question of mine, just with a different formulation of the identity. – rschwieb Jan 28 at 12:07
• @rschwieb,does this identity could be implied by Hua's identity? – yuan Jan 28 at 12:13
• I have added an answer to this question in math.stackexchange.com/questions/1602573/… – Jose Brox Jan 28 at 16:16