Before painstakingly defining all these terms, let me ask my question in plain english: given a $G$-bundle, is every conjugate of a vector field by a gauge transformation an element of the Lie algebra?

For a little more context, the question arises from a calculation in Exercise 86 of Part II (Gauge Fields) of *Gauge fields, knots, and gravity* by Baez and Muniain. Spurred on by the comments, I have decided to spell out the terminology in full detail as follows.

Let $M$ be a smooth manifold, let $G$ be a Lie group, and let $ E\to M$ be a $G$-bundle, viz. a vector bundle whose model fiber $V$ is a (finite dimensional) vector space equipped with a representation $\rho\colon G\to \mathrm{GL}(V)$ such that all bundle transition maps belong to $\rho(G)$. Denote the endormorphism bundle of $E$ by $\mathrm{End}(E)$ and let $G(E)\subset \Gamma\bigl(\mathrm{End}(E)\bigr)$ denote the gauge group of the bundle. Elements $g\in G(E)$ are sections of $\mathrm{End}(E)$ such that for every local trivialization $\phi\colon \mathrm{End}(E)\restriction U\to U\times \mathrm{GL}(V)$ and for every $p\in U$, it holds that $\phi(g_p)\in \{p\}\times \rho(G)$.
The derivative $d\rho\colon\mathfrak{g}\to \mathrm{End}(V)$ furnishes an embedding of the Lie algebra of $G$ into the set of linear transformations of $V$. Paralleling the definition of $G(E)$, we define the *infinitesimal* gauge group $\mathfrak{g}(E)$ to be the set of sections $\mathcal{G}\in\mathrm{End}(E)$ such that for $\phi(\mathcal{G}_p)\in \{p\}\times d\rho(\mathfrak{g})$ for every $p$ and for every local trivialization $\phi$ around $p$. Finally, on any trivialization $\phi$ of $\mathrm{End}(E)\restriction U$ let $D^{\phi}$ denote the standard flat connection on the trivialization.

**Question.**
For all vector fields $X$ and $\rho,\phi, U$ as above is $\rho(g)^{-1}D^\phi_X\rho(g)\in \mathfrak{g}(E\restriction U)$?

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