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, leaving
$$
\boxed{w^r=(C^r{}_{st}-C^r{}_{ts})x^sy^t+O(3)}.
$$
The antisymmetric part of the local multiplication law is therefore the infinitesimal <group commutator>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-302/2/iv/solution