Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 2 / iv

Insert the expansions from parts ii and iii successively into
$g(w)=g(x{)}^{-1}g(y{)}^{-1}g(x)g(y)$. All constant and linear terms cancel, leavingThe antisymmetric part of the local multiplication law is therefore the infinitesimal group commutator.