Directly commuting the vector fields gives
$$
[T_a,T_b]
=(\mu_a{}^s\partial_s\mu_b{}^r
-\mu_b{}^s\partial_s\mu_a{}^r)\partial_r.
$$
Using the definition in part x and $\lambda\mu=I$ yields
$$
\boxed{[T_a,T_b]=f^c{}_{ab}T_c}.
$$


Solution

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

 Solution

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