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Lets say that we know the value of the commutators [A,B] and [A,C]. Is there any way so that we can calculate value of commutator [B,C]? I looked up various sources for commutator identities, but somehow all of them fail to solve this.

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  • $\begingroup$ Well, if $A=0$ then $[A,B]=0=[A,C]$. That can't really help you to say anything about $[B,C]$... $\endgroup$ – Arnaud D. Mar 15 '18 at 13:20
  • $\begingroup$ What if A is not equal to zero. Can we say anything about [B,C]? $\endgroup$ – Jitendra Mar 15 '18 at 13:21
  • $\begingroup$ Why the commutative algebra tag? $\endgroup$ – Mohan Mar 15 '18 at 14:03
  • $\begingroup$ You should say that center of group doesn't contain $A$ $\endgroup$ – openspace Mar 15 '18 at 14:13
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If $A$ is in the centre of the group/Lie algebra (I don't know what the context is), then the first two commutators are trivial, while the third can be anything.

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  • $\begingroup$ First two are already known. $\endgroup$ – Jitendra Mar 15 '18 at 13:23

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