Write $g(x)=I+x^iT_i+O(x^2)$ in a <matrix representation> and $[T_i,T_j]=c^k{}_{ij}T_k$. Direct multiplication gives
$$
g(y)^{-1}g(x)^{-1}g(y)g(x)
=I+y^ix^j[T_i,T_j]+O(3),
$$
so
$$
w^k=c^k{}_{ij}y^ix^j+O(3)=-c^k{}_{ij}x^iy^j+O(3).
$$
The cancellation of the constant and linear terms explains why the <group commutator> first detects the <Lie bracket> at quadratic order.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / c / Solution

{title2=$[x,y]=x^{-1}y^{-1}xy$}
{wiki=Commutator}

The group commutator measures the failure of two elements to commute. Under the convention $[x,y]=x^{-1}y^{-1}xy$, it is the identity exactly when $xy=yx$.


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