Antisymmetrize part ix in $a,b$. The left side vanishes because the second derivative is symmetric in $s,t$. Therefore
$$
f^c{}_{ab}(y)=f^c{}_{ab}(z),
\qquad
f^c{}_{ab}
=[T_a\mu_b{}^r-T_b\mu_a{}^r]\lambda_r{}^c.
$$
Since multiplication by an arbitrary $g(x)$ moves $y$ to any $z$ in the connected coordinate neighborhood, the $f^c{}_{ab}$ are constants. They are the <structure constants>.

Solution

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

 Solution

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