nLab graded commutator

In superalgebra the the graded commutator or supercommutator of two elements of homogeneous degree is

$$[a,b] \coloneqq \left( a b - (-1)^{deg(a) deg(b)} b a\right)$$

and extended from there to the whole superalgebra as a graded derivation in both arguments.

So when at least one of $a$ or $b$ is even graded, then this is the commutator. When both are odd graded then this is the anti-commutator.

The graded commutator is just the plain commutator internal to the symmetric monoidal category of super vector spaces.