Solution
= Solution
In exponential coordinates, the <Baker--Campbell--Hausdorff formula> gives
$$
g(x)g(y)
=\exp\left\{(x^c+y^c)T_c
+\frac12x^ay^b[T_a,T_b]+O(3)\right\}.
$$
Comparison with $\varphi^c(x,y)=x^c+y^c+C^c{}_{ab}x^ay^b+O(3)$ gives
$$
\boxed{f^c{}_{ab}=C^c{}_{ab}-C^c{}_{ba}
=2C^c{}_{[ab]}}.
$$