Let $\pi:P \rightarrow M$ be a principal $G$-bundle, and let $A \in \mathfrak{g}$, where the Lie algebra of $G$ is indicated. The *fundamental field* $A$^{#} used to define connections is given by

$A$^{#}$(p) := \frac{d}{dt}(\exp(At)p)|_{t=0}$.

$A$^{#} is well defined since $e^{At}p$ can be regarded as a vector in $\pi^{-1}(\pi(p))$. Intuitively, I try to think of $A$^{#} as the (vertical) direction of the displacement on the fiber generated by $A$.

By the defining properties of principal bundles (in particular, the free action of $G$), we have $\{A$^{#} $: A \in \mathfrak{g}\} \simeq \mathfrak{g} \simeq \mathcal{V}(p)$, where $\mathcal{V}(p)$ is the vertical subspace at $p$. Something like

$A$^{#}$(p) = \lim_t t^{-1}(e^{At}p - p) = A \cdot p$

would be (to me) a nice way of thinking about $A$^{#}$(p)$, except that this is (at best) a formal equality. More precisely, $L_{\exp(At)}$ is the 1-parameter group of diffeomorphisms generated by $A$^{#}, where $L_g$ denotes left multiplication by $g$.

**My question**: is there a better way of thinking about a fundamental field than (either the definition itself) or the notional equation above?