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

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.

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 $$t$$.