Commutator Properties: [A,B]C+B[A,C]=[A,B](C+B)?

[Kyle Nemeth]

Given the property,

[A,BC] = [A,B]C+B[A,C],

is it true that, if B=C, then

[A,B]C+B[A,C]=[A,B]C+B[A,B]=[A,B](C+B)?

I apologize if I have posted in the wrong forum.

 

[fresh_42]

Given the property,

[A,BC] = [A,B]C+B[A,C],

is it true that, if B=C, then

[A,B]C+B[A,C]=[A,B]C+B[A,B]=[A,B](C+B)?

I apologize if I have posted in the wrong forum.

The commutator (in the assumed context above) is defined as $[A,B]=AB-BA$.
Now you have $[A,B]C+B[A,C]=[A,B]B+B[A,B]=AB^2-B^2A$ on the left and a factor $2$ on the right.
The (presumably) operators (or linear mappings) $A,B,C$ do not commutate, so you cannot switch orders.

 

[Kyle Nemeth]

Thank you for your help. I understand now.