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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.

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    $\begingroup$ 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. $\endgroup$ – quarague Jan 28 at 10:33
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    $\begingroup$ Conjugation in a group is not "obscure". $\endgroup$ – Dietrich Burde Jan 28 at 10:48
  • $\begingroup$ 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. $\endgroup$ – rschwieb Jan 28 at 12:07
  • $\begingroup$ @rschwieb,does this identity could be implied by Hua's identity? $\endgroup$ – yuan Jan 28 at 12:13
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    $\begingroup$ I have added an answer to this question in math.stackexchange.com/questions/1602573/… $\endgroup$ – Jose Brox Jan 28 at 16:16

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