Why don't I see "vector-valued vector fields"?

Question

Let $P\xrightarrow{\pi}M$ be a principal $G$-bundle with connection $\omega\in \Omega^1(P;\mathfrak{g})$. When I am studying these things, there are Lie-algebra valued differential forms all over. These should be some $\alpha:\Gamma(\bigwedge^kTP)\rightarrow C^\infty(P;\mathfrak{g})$.

But then I want to take Lie derivatives of these forms along vector fields on $P$ and I am confused about how to do this because my ordinary vector fields on $P$ are derivations on the algebra $C^\infty(P;\mathbb{R})$ and they satsify a Leibniz rule with respect to pointwise multiplication of $f,g\in C^\infty(M)$ $$X(p)(fg) = (X(p)f)g(p) + f(p)(X(p)g)$$

But e.g. here we want them to act on Lie-algebra valued functions, so it seems like we want derivations $X:C^\infty(P;\mathfrak{g})\rightarrow C^\infty(P;\mathfrak{g})$ that satisfy a different Leibniz rule with respect to the pointwise Lie algebra multiplication of $f,g\in C^\infty(P;\mathfrak{g})$. $$X[f,g] = [Xf,g]+[f,Xg]$$

Yet it's the same vector fields?

So my questions are

1. How does $\text{Der}(C^\infty(M))$ relate to $\text{Der}(C^\infty(P;\mathfrak{g}))$

2. Is this second Leibniz rule for Lie-algebra valued functions (such as $\omega(X)$) true and why would it be?

Also any explication of the theory or references for what I should make (geometrically or otherwise) of derivations of smooth functions with values in some other algebra would be handy.

Answer 1

The Lie derivative of any tensor (even if vector valued) is defined using a vector field and its flow. $$\mathcal L_X T := \left.\frac{d}{dt}\right\rvert_{t=0}\phi_t^*T$$ where $\phi_t:M\to M$ is the flow diffeomorphism associated to $X$ by time $t$.

This is done without reference to the action of a vector (at a point) as a derivation on the algebra of functions.

I'm not sure about the second equation you write being true. It might be.