I am reading Natural and Gauge Natural Formalism for Classical Field Theory by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.

Let's say we have a principal bundle $\mathcal{P}=(P, M, \pi ; G)$ and the isomorphism $T_{e} L_{p}: \mathfrak{g} \longrightarrow V_{p}(\pi): T_{A} \mapsto \lambda_{A}(p)$. He fixes a point $p=[x, g]_{(\alpha)}$. $\theta_{(L)}^{A}=\bar{L}_{a}^{A}(g) \mathrm{d} g^{a}$ is the local basis of left invariant 1-forms dual to $\lambda_{A}=L_{A}^{a}(g) \partial_{a}$ where the two matrices are inverse of each other. Then the connection one form can be written as \begin{equation} \bar{\omega}_{p}=\left[\theta_{(L)}^{A}(p)+\mathrm{Ad}_{B}^{A}\left(g^{-1}\right) \omega_{\mu}^{B}(x) \mathrm{d} x^{\mu}\right] \otimes T_{A} \end{equation} Pulling back with a section, he gets \begin{equation}\sigma^{*} \bar{\omega}=\left[\bar{L}_{a}^{A}(g) \partial_{\mu} g^{a}(x)+\mathrm{Ad}_{B}^{A}\left(g^{-1}\right) \omega_{\mu}^{B}(x)\right] \mathrm{d} x^{\mu} \otimes T_{A} \end{equation} However, then he goes on saying "We remark that the local gauge $\sigma$ also induces a local trivialization of $\mathcal{P}$. In the induces local trivialization, the section $\sigma$ has the expression $\sigma: x^{\mu} \mapsto\left(x^{\mu}, e\right)$ and the vector potential is of the form"

 \begin{equation} \sigma^{*} \bar{\omega}=\omega_{\mu}^{A}(x) \mathrm{d} x^{\mu} \otimes T_{A} \end{equation} He also states that the induces connection is of the form \begin{equation} \omega=\mathrm{d} x^{\mu} \otimes\left(\partial_{\mu}-\omega_{\mu}^{A}(x) \rho_{A}\right) \end{equation}

1. My first question is how one can derive the expression of $\bar{\omega}_{p}$?
2. My section question is why every other book on the subject I know uses $\sigma^{*} \bar{\omega}=\omega_{\mu}^{A}(x) \mathrm{d} x^{\mu} \otimes T_{A}$ as the definition of a form in local coordinates even this seems not to be true for a general section.
3. I am also not sure how one can derive the form of the induced connection.

Cross posted: https://math.stackexchange.com/questions/4128191/local-coordinates-of-one-form-on-a-principal-bundle

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.

Let's say we have a principal bundle $\mathcal{P}=(P, M, \pi ; G)$ and the isomorphism $T_{e} L_{p}: \mathfrak{g} \longrightarrow V_{p}(\pi): T_{A} \mapsto \lambda_{A}(p)$. He fixes a point $p=[x, g]_{(\alpha)}$. $\theta_{(L)}^{A}=\bar{L}_{a}^{A}(g) \mathrm{d} g^{a}$ is a local basis of left invariant 1-forms dual to $\lambda_{A}=L_{A}^{a}(g) \partial_{a}$ where the two matrices are inverses of each other. Then the connection 1-form can be written as \begin{equation} \bar{\omega}_{p}=\left[\theta_{(L)}^{A}(p)+\mathrm{Ad}_{B}^{A}\left(g^{-1}\right) \omega_{\mu}^{B}(x) \mathrm{d} x^{\mu}\right] \otimes T_{A}. \end{equation} Pulling back with a section, he gets \begin{equation}\sigma^{*} \bar{\omega}=\left[\bar{L}_{a}^{A}(g) \partial_{\mu} g^{a}(x)+\mathrm{Ad}_{B}^{A}\left(g^{-1}\right) \omega_{\mu}^{B}(x)\right] \mathrm{d} x^{\mu} \otimes T_{A}. \end{equation} However, then he goes on saying "We remark that the local gauge $\sigma$ also induces a local trivialization of $\mathcal{P}$. In the induced local trivialization, the section $\sigma$ has the expression $\sigma: x^{\mu} \mapsto\left(x^{\mu}, e\right)$ and the vector potential is of the form"  \begin{equation} \sigma^{*} \bar{\omega}=\omega_{\mu}^{A}(x) \mathrm{d} x^{\mu} \otimes T_{A} \end{equation} He also states that the induces connection is of the form \begin{equation} \omega=\mathrm{d} x^{\mu} \otimes\left(\partial_{\mu}-\omega_{\mu}^{A}(x) \rho_{A}\right) \end{equation}

1. My first question is how one can derive the expression for $\bar{\omega}_{p}$?
2. My second question is why every other book on the subject I know uses $\sigma^{*} \bar{\omega}=\omega_{\mu}^{A}(x) \mathrm{d} x^{\mu} \otimes T_{A}$ as the definition of a form in local coordinates; even this seems not to be true for a general section.
3. I am also not sure how one can derive the form of the induced connection.

Cross posted: https://math.stackexchange.com/questions/4128191/local-coordinates-of-one-form-on-a-principal-bundle