Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. If $(U, \phi)$ is a local trivialization of this bundle then for every $x\in M$ we have a diffeomorphism $$\phi_x:P_{x}\longrightarrow G, \phi_x:=\textrm{pr}_2\phi|_{P_x},$$ where $P_x:=\pi^{-1}(x)$. We might then induce an action of $G$ on $P$ as follows $$p\cdot g:=\phi_{\pi(p)}^{-1}(\phi_{\pi(p)}(p)g).$$ That is not difficult to see:
$(i)$ This action does not depend on the choice of the trivialization;
$(ii)$ $\pi(p\cdot g)=\pi(p)$ and therefore the action preserves fibers.
$(iii)$ If $\tau_g:P\longrightarrow P$, $p\longmapsto p\cdot g$ then $(d\tau_g)_p(V_p)\subset V_p$ where $V_p:=T_p P_{\pi(p)}$ is the subspace of $T_pP$ of vertical vectors.
How can I construct a $G$-left invariant vector field using this data?
I'm following Lichnerowicz's book Théorie globale des connexions et des groupes d'holonomie and since it is a bit old I can't follow the argument he presents.
He goes as follows (in an up-to-date notation):
$(i)$ If $z\in P_x$ then $z=\phi_x^{-1}(\gamma)$ for some $\gamma\in G$. But $$z=\psi_x^{-1}(\psi_x\phi_x^{-1}(\gamma))=\psi_x^{-1}(g\gamma),$$ where $(V, \psi)$ is another trivialization such that $x\in U\cap V$.
$(ii)$ $\tau_g$ preserves fibers and vertical vectors;
$(iii)$ If $\tau$ is a vertical vector in $z$ and $z=\phi_x^{-1}(\gamma)$ then we might associate a tangent vector $\theta_\gamma\in T_\gamma G$.
$(iv)$ From this vector, using $$\theta_{g\gamma}=g\theta_\gamma$$ we find a $G$-left invariant vector field.
I think $(i)-(iv)$ has to do with the identity $$(dL_g)_\gamma\circ (d\phi_x)_z=(d\psi_x)_z\circ (d\tau_g)_z$$ although I'm not very sure.
Any help will be welcome.