I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis.
Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or the Lie bracket of two basis vectors, i.e. $\mathcal{L}_{e_a}e_b=C^c_{ab}e_c$.
The expression I got down to was $$\mathcal{L}_{e_a}g= e_a^{\mu}\partial_{\mu}{g_{cb}}\ e^c\otimes e^b\ + C^c_{da}g_{cb}\ e^d\otimes e^b \ + C^b_{da}g_{cb} e^c \otimes e^d.$$ I was thinking this is pretty similar to what the covariant derivative is for a metric, but with the usual connection replaced with the Cs, and + signs where we'd usually have -. However, I don't see why the Cs need to be constant in order for this to be true. In an earlier part of the question we showed that if the C are constant they are the structure constants of the Lie algebra formed by the non-coordinate basis. I think this earlier part is given as a hint probably, but I'm unsure of how to use it.
I was also wondering if the derivative of the metric here would be zero? In our notes it says any non-coordinate basis we work with will be an orthonormal one. So the metric is just the usual minkowski, is it not? And it is a scalar with respect to the spacetime coordinates.
