Properties of anticommutators Do anticommutators of operators has simple relations like commutators.
For example: 
$$[AB,C]=A[B,C]-[C,A]B.$$
But I don't find any properties on anticommutators. Do same kind of relations exists for anticommutators?
 A: As you can see from the relation between commutators and anticommutators
$$
[A,B] := AB-BA = AB - BA -BA + BA = AB + BA - 2BA = \{A,B\} - 2 BA
$$
it is easy to translate any commutator identity you like into the respective anticommutator identity. Unfortunately, you won't be able to get rid of the "ugly" additional term.
This is probably the reason why the identities for the anticommutator aren't listed anywhere - they simply aren't that nice.
A: First, let's prove the result you gave
$$
[AB,C] = ABC-CAB = ABC-ACB+ACB-CAB = A[B,C] + [A,C]B.
$$
Notice that $ACB-ACB = 0$, which is why we were allowed to insert this after the second equals sign.
Now let's consider the equivalent anti-commutator $\lbrace AB , C\rbrace$; using the same trick as before we find
$$
\lbrace AB,C \rbrace = ABC+CAB = ABC-ACB+ACB+CAB = A[B,C] + \lbrace A,C\rbrace B
$$
Here are a few more identities from Wikipedia involving the anti-commutator that are just as simple to prove:
$$
{\displaystyle \{A,BC\}=\{A,B\}C-B[A,C]}
$$
$$
{\displaystyle \{AB,C\}=A\{B,C\}-[A,C]B}
$$
$$
{\displaystyle [AB,C]=A\{B,C\}-\{A,C\}B}
$$
https://en.wikipedia.org/wiki/Commutator#Identities_.28ring_theory.29
Hope this helps.
