# minor things

1) There is no need to introduce notation F for the field. Simply say
Let $A$ be a not-necessarily associative algebra over a field.
or
Let $A$ be a not-necessarily associative algebra.

2) 'An associator' implies that there are many associators. This
is not in line with the definition. What about:

The mapping $[\ , , ] \colon A\times A\times A \to A$ given by
$$[a,b,c]=(ab)c-a(bc),\quad a,b,c\in A$$
is the \emph{associator} in $A$.

3) Consider:
'$[ , , ]$ is trivial' -> '$[\ , , ]$ is identically zero'
or 'the associator vanishes identically'

This is more precice.