Connection on principal $G$-bundle v.s. connection on associated vector bundle I'm confused with the relation between these two. I'll summarize what I know so far and explain my questions.
So for a principal $G$-bundle $P$, we can define a principal $G$-connection on it, we can then have an induced connection on the associated fiber bundle $F$. Also, this connection is a 1-form with values in the Lie algebra $\mathfrak{g}$ of $G$.
On the other hand, we have for a vector bundle $V$ with structure group $G$, we can then have a subbundle of its frame bundle $F(V)$, which is a principal $G$-bundle. Also, this $G$-structure on the vector bundle is usually associated to some extra structure. For example, if $G=O(n)$, that means we have a fiber metric.
I wonder if the following statements are true:

*

*For a connection on the $V$, can we induce from it a principal $G$-connection on the $F(V)$? If this is not always the case, can we put a little extra structure on the connection that could make this true? e.g. to make the connection compactible with the $G$-structure on the vector bundle. (But I only know how to describe this compactibility for groups like $O(n),U(n)$)


*For the 1-form with values in $g$ that represents the principal $G$-connection on $P$, what does the induced connection on $V$ look like? Since a connection on a vector bundle is always represented by a matrix of 1-form, do we always have the 1-form matrix takes value in $\mathfrak{g}$  (i.e. that is to say connection matrix itself as a matrix is in $\mathfrak{g}$)
P.S. Any help is a ppreciated. Also, I did not find some good reference about this topic, so please let me know if you know some good places that can answer these questions.
 A: So the answer to (1.) is positive. If you have a connection on a vector bundle you also obtain a connection form on the associated frame bundle.
In general you need to work a bit to see this. If $\mathcal{A}$ is a connection form on a principal $G$-bundle $P\to M$, then for each local section $s:U\to P$, $U\subset M$ in $P$ you can locally represent it as $s^*\mathcal{A}\in\Omega^1(U,\mathfrak{g})$. By chosing a family of sections whose domains cover $M$ you have now stored all information on $\mathcal{A}$ in these local forms and can reconstruct $\mathcal{A}$ from these. Moreover, they are subdued to a certain transformation formula if you change the section,
$$
s_i^*\mathcal{A}=\text{Ad}_{g_{ij}^{-1}}\circ s_j^*\mathcal{A}+g_{ij}^*\,\mu^{\text{MC}},
$$
with the Maurer-Cartan form on $G$ and the functions $g_{ij}:U_i\cap U_j\to G$ which translate the sections $s_i:U_i\to P$ and $s_j:U_j\to P$ into one another on their common domain.
Now do the converse, whenever you have a family of $\mathfrak{g}$-valued 1-forms $\{\mathcal{A}_i\in\Omega(U_i;\mathfrak{g})\}_i$, whose domains $U_i$ cover the basis $M$ and a family of local sections $\{s_i:U_i\to P\}_i$ such that the above formula holds, then you can built a connection form $\mathcal{A}$ on all of $P$ such that $s_i^*\mathcal{A}=\mathcal{A}_i$.
If you now start with a vector bundle $V\to M$ with fiber type $V_0$ and a connection $\nabla$ thereon, you can choose a local frame $e_1,\dots,e_n:U\to V$ and represent the action of $\nabla$ as a matrix of 1-forms, $\nabla e_i=\sum_j \omega_{ji}\otimes s_j$. That way you can write $\omega$ as a 1-form with values in $\text{End}(V)$. The endomorphism bundle $\text{End}(V)$ is now of fiber type $\text{End}(V_0)$, which is exactly the Lie algebra of the frame bundle's structure group $\text{GL}(V_0)$. If you had additional structure, e.g. a fiber metric, you would have chosen an orthonormal frame, would have received a skew symmetric matrix which in in the Lie algebra of $\text{O}(n)$, and so on.
For these Lie-algebra valued forms you can now show the following: The chosen frame for finding the form $\omega$ defines a section in the frame bundle. If you now cover your whole basis $M\subset\bigcup_iU_i$ with domains of local frames of $V$, or sections of its frame bundle, respectively, then you find that the above forms fulfill the required transformation law from above, and thus you can build them together in a connection form on the frame bundle (or subbundles with a smaller structure group in the same way).
For (2.) you also need to work a bit, and I will not work that out in detail. But, say you have a principal $G$-bundle $P\to M$ and an associated vector bundle $V:=P\times_{\rho,G} V_0\to M$ via a representation $\rho$ of $G$ on $V_0$, then the crucial ingredient is an isomorphism between $\Omega_\text{hor}(P;V_0)^{(G,\rho)}$ and $\Omega(M,V)$. Hereby, $\Omega_\text{hor}(P;V_0)^{(G,\rho)}$ is the space of differential forms on $P$ with values in $V_0$, which transform along the fibers of $P$ via the representation $\rho$, and which are horizontal.
By means of this isomorphism you can push back and forth any action on commensurable $V_0$-values forms on $P$ to $V$-valued forms on $M$, particularly you can push the horizontal derivative on horizontal forms to $V$-valued forms on $M$ and easily show that this action defines a covariant derivative on $V$ (sections in $V$ are just $V$-valued 0-forms). This association is again "inverse" to the above in the sense that, if you have started with $V$ and $\nabla$ thereon as above, and let $\mathcal{A}^\nabla$ be the associated connection form on the frame bundle of $V$, then what I just described reproduces $\nabla$ on $V\cong P\times_{(G;\text{id})}V_0$ (with the trivial representation $\text{id}:\text{GL}(V_0)\to\text{GL}(V_0)$).
But on one thing you need to watch out a bit. While a connection form on a principal bundle is a Lie algebra valued form in general (by definition), a connection on a vector bundle is not, only locally in a frame as I described above. So I think keeping in mind that "connections on vector bundles are Lie algebra valued forms" is a bit misleading, since if they were globally such forms, that would kind of kill their property as a way of covariantly differentiating.
I hope I could help you a bit, though the second part might be a bit large or general of a question for MSE (you can white whole math book chapters on this topic).
Since you asked, I liked the book of Helga Baum. At some points it may be hard to follow, but I think she develops a good viewpoint on all these things. I think it's only available in German, but I also found various lecture notes in English which are based on this book on google (but forgot who the authors were).
