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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-302/1/c/solution