dg.differential geometry - What is a good way to think about a fundamental field on a principal G-bundle?

I find that the notion of fundamental vector field is well defined not only for $G$-principal bundle but even for any $G$-manifold, i.e. a manifold with an action by a Lie group $G$.

About your notional equation, I would say that the fundamental vector fields effectively arise from an action of $frak{g}$ on $M$. However some clarifications are needed.

Let $Psi:Mtimes Gto M$ be a right action of a Lie group $G$ on a manifold $M$.
Let $frak{g}$ be the Lie algebra of $G$, viewed as formed by the left invariant vectorfields on $G$.

Then there exists a unique map $zeta^{Psi}equivzeta:Xinfrak{g} mapsto$$zeta_Xin mathfrak{X}$$(M)$ such that $(Tpsi)circ(0_M+X)=zeta_XcircPsi$, $zeta_X$ and $0_M+X$ are $Psi$-related, for any $Xinfrak{g}$.(Above $0_M$ denoted the zero vectorfield on $M$.)
For any $Xinfrak{g}$, the vector field $zeta_X$ on $M$ is called the fundamental vectorfield corresponding to $X$ w.r.t. the right action $Psi$.

The definition of $zeta$ is well posed just because, for any $Xinfrak{g}$, the map $TPsicirc(0_M+X)$ is constant on the fibers of $Psi$; and this holds being $Psi$ a right action and $X$ a left invariant vectorfield.

Obviously the following properties are satisfied:

• $zeta_{aX+bY}=azeta_X+bzeta_Y,zeta_{[X,Y]}=[zeta_X,zeta_Y]$, for any $a,binmathbb{R}$, and $X,Yinfrak{g}$, i.e. $zeta:mathfrak{g} to mathfrak{X} (M)$ is a Lie algebra homomorphism;


• $zeta_X$ is complete and its $t$-time flow is $Psi^{exp{tX}}$, for any $Xinfrak{g}$ and $tinmathbb{R}$.

For an abstract Lie algebra $frak{g}$, an action of $frak{g}$ on a manifold $M$ is defined to be a Lie algebra homomorphism from $frak{g}$ to $frak{X}$$(M)$.
In such a way for any right action $Psi$ of a Lie group $G$ on $M$, we have that $zeta^{Psi}$ is an action on $M$ by the Lie algebra of $G$.

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