# Commutator Algebra

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.

• Well, if $A=0$ then $[A,B]=0=[A,C]$. That can't really help you to say anything about $[B,C]$... – Arnaud D. Mar 15 '18 at 13:20
• What if A is not equal to zero. Can we say anything about [B,C]? – Jitendra Mar 15 '18 at 13:21
• Why the commutative algebra tag? – Mohan Mar 15 '18 at 14:03
• You should say that center of group doesn't contain $A$ – openspace Mar 15 '18 at 14:13

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.