associator

Let $A$ be a non-associative algebra over a field. The associator of $A$, denoted by $[,,]$, is a trilinear (http://planetmath.org/multilinear) map from $A\times A\times A$ to $A$ given by:

$$[a,b,c]=(ab)c-a(bc).$$

Just as the commutator measures how close an algebra is to being commutative, the associator measures how close it is to being associative. $[,,]=0$ identically iff $A$ is associative.