I'm a bit confused about Lie derivative on principal bundle $P(M,G)$.
Let $g(t)$ be the flow generated by vector fields $Y$ and $X$ also vector field on $P$. According to the definition, $$\mathcal{L}_YX = \lim_{t\to0} \frac{g_{-t*}X|_t-X|_0}{t}$$ But the book claims that $$\mathcal{L}_YX= \lim_{t\to0} \frac{R_{g(t)*} X-X}{t}$$ where $R_{g(t)}$ is the right translation action.
Then I am confused why $g_{-t*}X|_t=R_{g(t)*} X|$? Relevant facts that I can recall is push forward of vectors is just matrix multiplication $f_* V=fV$ for Lie group. But why right translation not left translation here? And why is not $g(-t)$ but $g(t)$?
I'm a beginner and probably read too fast in the Lie derivative section. Any help would be very greatly appreciated!