Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. Let $\mathfrak{g}$ be the Lie algebra of left-invariant vector fields on $G$ and $\mathfrak{V}(P)$ be the space of vertical vector fields on $P$, that is: $$\mathfrak{V}(P):=\{X\in\mathfrak{X}(P): X_p\in T_p P_{\pi(p)}\},$$ where $\mathfrak{X}(P)$ stands for the space of vector fields on $P$.
There are two ways to define a map $$\mathfrak{g}\longrightarrow \mathfrak{V}(P), X\longmapsto \sigma(X),$$ such that $$\mathfrak{g}\longrightarrow V_p, X\longmapsto \sigma(X)_p,$$ is a linear isomorphism for every $p\in P$ where $V_p:=T_pP_{\pi(p)}=\textrm{Ker}(d\pi_p)$.
First Way: Take $(U, \phi)$ a local trivialization with $\pi(p)\in U$ and set $$\sigma(X)_p:=(d\phi_{\pi(p)}^{-1})_{\phi_{\pi(p)}(p)}(X_{\phi_{\pi(p)}(p)}),$$ where $\phi_{\pi(p)}:P_{\pi(p)}\longrightarrow G$ is the diffeomorphism associated to $\phi$, that is, $\phi_{\pi(p)}:=\textrm{pr}_2\circ \phi|_{P_{\pi(p)}}$. Of course, it is possible to show this map does not depend on the choice of local trivialization.
Second Way: Given $X\in\mathfrak{g}$ one defines $\sigma(X)\in \mathfrak{V}(P)$ setting $$\sigma(X)_p:=\frac{d}{dt}(L_p\circ \exp(tX))\biggr|_{t=0},$$ where $L_p:G\longrightarrow P$, $g\longmapsto p\cdot g$.
Can anyone explain-me the relation between those constructions?
Thanks.
Remark. The action of $G$ on $P$ is given by $$p\cdot g=\phi_{\pi(p)}^{-1}(\phi_{\pi(p)}(p)g).$$