I am wondering if it is possible do extend Cartan's magical formula for complex valued smooth forms. To be more specific I want do know the answer to the following questions:
Let $M$ be a smooth manifold, $X$ a real-valued smooth vector field and a smooth $k$-form $\omega : \Lambda^k \mathbb{C}TM \to \mathbb{C}$. I want to know if the two following formulas are true: $$ L_X w = d(i_X w) + i_X (d w) $$ in which $L_X$ is the Lie derivative and $i_X$ is the retraction.
I also want to know if, given two smooth forms $u,v$, it holds that: $$ L_X (u \wedge v) = (L_X u) \wedge v + (-1)^\sigma (L_X v).$$
It seems that it is possible to achieve the above formulas just by linear extension of the real versions of them, but I don't want to write all the computations down just to be sure, so I'm asking if they are really true and if they are, where can I find a reference for them?
Thank you very much.