Given a principal $G$ bundle $E\to M$, where $G$ is a Lie group, I was told that for every $u \in \mathfrak{g}$ (the Lie algebra of $G$) we can define a $G$-invariant vector field $X_u$ over $E$. I'm trying to figure out how this field is defined:
since we have canonical identifications of $\mathfrak{g}\cong T_gG$ for every $g$, it makes sense to expect that the vector field will associate to each point in a fiber, the vector represented by $u$ (seen as a vector in $T_gG$). The problem is that this association seems to depend on how I identify the fibre with $G$ and I'm not able to prove that it's independent from it (hence well-defined).
Assuming it's well-defined, it's clearly left-invariant (I hope it's what it's meant by $G$-invariant).
How can I show that's well-defined?
EDIT: the statement above comes from "John Roe's Elliptict Operators, Topology and asymptotic methods" page 23, at the bottom
If $p:E\rightarrow M$ is a principal $G$-bundle, $E$ is endowed with an action of $G$ whose quotient space is $M$, let $u\in{\cal G}$, define $X_u(x) = \left. \frac{d}{dt} \right|_{t=o} \exp(tu) \mathbin{.} x$.