Im working on some things in Gauge Theory and have some problems proving a rather simple result. For any Lie Group $G$ acting on a manifold $M$, one can define the fundamental vector field of $X$ on $M$, defined by differentiating the group action $\rho: G \times M \longrightarrow M $:
$$ X^*_p := \frac{\text d}{\text dt}\Big|_{t=0} \rho(\exp(tX), p) $$
I have shown that for right actions one has $[X^*,Y^*] = [X,Y]^*$, what remains to show is $[X^*,Y^*] = -[X,Y]^*$ for left actions. However, the sign always seems to cancel out. Can somebody help me out? Thanks in advance.
You can find a proof in many books on Lie groups. I would refer you to Symplectic geometry and analytical mechanics, by Libermann and Marle, but there are equivalent sources.
In particular, the computation you are looking for should be in the proof of Proposition 3.8 in appendix 5. Notice that there they work with left actions, see Definition 3.6 for their convention on infinitesimal generators of Lie groups actions.
Hope this helps!