What is magical about Cartan's magic formula?

Solution 1:

I would have liked to put this as a comment but I do not have enough points to do it.

I think there is some magic in this formula because it tells us that the Lie derivative $\mathscr{L}_X$ is homotopic to zero with the homotopy $i_X$ going from top-right $\Omega^{p+1}(M)$ to bottom-left $\Omega^p(M)$ diagonally in the following diagram:

$$\require{AMScd} \begin{CD} \cdots @>{d}>> \Omega^p(M) @>{d}>> \Omega^{p+1}(M) @>{d}>> \cdots \\ \qquad @V{\mathscr{L}_X}V{0}V @V{\mathscr{L}_X}V{0}V\\ \cdots @>{d}>> \Omega^p(M) @>{d}>> \Omega^{p+1}(M) @>{d}>> \cdots \end{CD}$$