How to prove a property of Lie derivatives

Question

I know that there are five properties for Lie derivative. But one of them I don't know how to prove. It is $ L_x[\omega(Y)]=(L_x \omega)(Y)+\omega(L_x Y)$

Note : Here $\omega$ is a covariant vector field while Y is a vector field. Is there anyone has any idea?

Answer 1

I'll prove this as a collection of exercises. All of the steps should be quite doable from the definition.

Exercise A: prove that $\mathcal{L}_X Y = [X,Y]$

Exercise B: prove that $\mathcal{L}_X\omega = d(X\lrcorner\omega)+X\lrcorner d\omega$

Exercise C: prove that $d\omega(X,Y) = X(\omega(Y))-Y(\omega(X))-\omega([X,Y])$

Now put these all together (remembering the Leibniz property of the Lie derivative).