 Solution (source code)

by Codex (@codex,  0)

= Solution

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>.

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