I actually try to read the Geometry, Topology and Physics by M.Nakahara and I'm stuck on the Lie derivative of a r-form and the demonstration of $i_{[X,Y]}\omega = [L_X,i_Y]\omega $ . I found some proof of this relation like in this pdf https://www.mathi.uni-heidelberg.de/~lee/Xiaoman_Wu.pdf . But this use a different definition of the Lie derivative than the Nakahara :
$L_X \omega =\frac{1}{r!} X^\nu\partial_\nu\omega_{\mu_1 \dots\mu_r}dx^{\mu_1}\wedge \dots \wedge dx^{\mu_r} + \frac{1}{r!}\sum_{s=1}^r \partial_{\mu_s}X^\nu\omega_{\mu_1 \dots\nu\dots\mu_r}dx^{\mu_1}\wedge \dots \wedge dx^{\mu_r}$
in Nakahara become :
$(L_X \omega)(X_1,\dots,X_r) = X[\omega(X_1,\dots,X_r)] -\sum_{i=1}^r \omega(X_1,\dots,[X,X_i],\dots,X_r) $
in the pdf.
I can't find how to connect this 2 definitions, thanks for help
I think I've fund the solution : $$\begin{align} (L_X w)(X_1\dots X_r)&=\frac{1}{r!}X^\nu \partial_\nu w_{\mu_1 \dots \mu_r}dx^{\mu_1} \wedge \dots \wedge dx^{\mu_r} (X_1 \dots X_r) + \frac{1}{r!}\sum_{s=1}^r \partial_{\mu_s} X^\nu w_{\mu_1\dots\nu\dots\mu_r}dx^{\mu_1}\wedge \dots\wedge dx^{\mu_s}\wedge\dots \wedge dx^{\mu_r} (X_1 \dots X_r)\\ &=\frac{1}{r!}X^\nu \partial_\nu w_{\mu_1 \dots \mu_r}\sum_{\sigma \in P} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(r)}^{\mu_r} + \frac{1}{r!}\sum_{s=1}^r \partial_{\mu_s} X^\nu w_{\mu_1\dots\nu\dots\mu_r}\sum_{\sigma \in P} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(s)}^{\mu_s} \dots X_{\sigma(r)}^{\mu_r}\\ &=\frac{1}{r!}X^\nu \partial_\nu \left[\sum_{\sigma \in P}w_{\mu_1 \dots \mu_r} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(r)}^{\mu_r}\right] - \frac{1}{r!}X^\nu w_{\mu_1 \dots \mu_r} \partial_\nu \left[\sum_{\sigma \in P} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(r)}^{\mu_r}\right] + \frac{1}{r!}\sum_{s=1}^r \partial_{\mu_s} X^\nu w_{\mu_1\dots\nu\dots\mu_r}\sum_{\sigma \in P} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(s)}^{\mu_s} \dots X_{\sigma(r)}^{\mu_r}\\ &=X\left[w(X_1,\dots, X_r)\right] - \frac{1}{r!}X^\nu w_{\mu_1 \dots\mu_s\dots \mu_r} \sum_{\sigma \in P} \text{sgn}(\sigma)\sum_{s=1}^r X_{\sigma(1)}^{\mu_1} \dots\partial_\nu X_{\sigma(s)}^{\mu_s}\dots X_{\sigma(r)}^{\mu_r} + \frac{1}{r!}\sum_{s=1}^r \partial_{\mu_s} X^\nu w_{\mu_1\dots\nu\dots\mu_r}\sum_{\sigma \in P} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(s)}^{\mu_s} \dots X_{\sigma(r)}^{\mu_r}\\ &=X\left[w(X_1,\dots, X_r)\right] - \frac{1}{r!}X^\nu w_{\mu_1 \dots\mu\dots \mu_r} \sum_{\sigma \in P} \text{sgn}(\sigma)\sum_{s=1}^r X_{\sigma(1)}^{\mu_1} \dots\partial_\nu X_{\sigma(s)}^{\mu_s}\dots X_{\sigma(r)}^{\mu_r} + \frac{1}{r!}\sum_{s=1}^r \partial_{\nu} X^{\mu_s} w_{\mu_1\dots\mu_s\dots\mu_r}\sum_{\sigma \in P} \text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1} \dots X_{\sigma(s)}^{\nu} \dots X_{\sigma(r)}^{\mu_r}\\ &= X\left[w(X_1,\dots, X_r)\right] - \frac{1}{r!} w_{\mu_1\dots\mu_s\dots\mu_r}\sum_{s=1}^r\sum_{\sigma \in P}\text{sgn}(\sigma)X_{\sigma(1)}^{\mu_1}\dots \left(X^\nu\partial_\nu X_{\sigma(s)}^{\mu_s}-X_{\sigma(s)}^{\nu}\partial_{\nu}X^{\mu_s}\right) \dots X_{\sigma(r)}^{\mu_r}\\ &= X\left[w(X_1\dots X_r)\right] - \sum_{s=1}^r w(X_1,\dots, [X,X_s], \dots,X_r) \end{align}$$
I hope there aren't too many mistakes