We know that given a group $G$, for every $x,y\in G$ we define $[x,y]:=x^{-1}y^{-1}xy$, the commutator of $x$ and $y$.
I saw something more general, commutators involving more than two elements, like $[x,y,z]$ (again all three elements in the same group $G$). How can we define them?
I suppose we have to use the definition above for the commutators of two elements; a generalized definition should naturally be coherent with the "base" definition.
But, for example if we ask that "the brackets have to behave freely" (and that should be a natural request) we define $[x,y,z]:=[[x,y],z]$; but this is not a good definition since in general $[[x,y],z]\neq[x,[y,z]]$ (it's just a computation).
What is the correct definition? How the brackets behave/has to behave?
Thanks a lot
$[x,y,z]=[[x,y],z]$ is the standard “left normed bracket convention” used with the definition $[x,y] = x^{-1} y^{-1} xy$. See for instance Robinson's textbook for a Course in the Theory of Groups section 5.1 page 119 in the 1ed, Gorenstein's Finite Groups section 2.2 page 19, etc.