For the definitions see:
Lie derivative for tensors in general,
In what follows: $X,Y$ vector fields, $\omega,\theta$ differential forms, not necessarily of the same degree.
1. $d \mathcal{L}_{X} \omega=\mathcal{L}_{X}(d \omega)$
2. $\mathcal{L}_{f X} \omega=f \mathcal{L}_{X} \omega+d f \wedge i_{X} \omega$ (only 1-forms?)
3. $\mathcal{L}_{X}(\omega\wedge \theta)=\mathcal{L}_{X}(\omega)\wedge \theta+\omega\wedge\mathcal{L}_{X}(\theta)$ (including functions)
4. $i_{[X, Y]}=\left[\mathcal{L}_{X}, i_{Y}\right]$ (warning: informal notation). It is the same as equation (1.62) in @olver86: $\mathbf{v}(\mathbf{w} \lrcorner \omega)=[\mathbf{v}, \mathbf{w}] \lrcorner \omega+\mathbf{w} \lrcorner \mathbf{v}(\omega)$. It is related to the following cases:
1. For a 1-form: $\mathcal{L}_X(i_Y \omega)=i_Y \mathcal{L}_X(\omega)+i_{[X,Y]}\omega$ , and with other notation
2. $\mathcal{L}_X(\langle \omega; Y\rangle)=\langle \mathcal{L}_X\omega;Y \rangle+\langle \omega;\mathcal{L}_X Y \rangle$ or, in general for a $k$-form,
3. $\mathcal{L}_X(\langle \omega; Y_1,\ldots,Y_k\rangle)=\langle \mathcal{L}_X \omega;Y_1,\ldots,Y_k \rangle+\langle \omega; \mathcal{L}_X Y_1,\ldots,Y_k\rangle+\ldots$. See @olver86 page 74 exercise 1.35.
5. infinitesimal Stokes' theorem
7. $\mathcal{L}_{X} \circ \mathcal{L}_{Y}-\mathcal{L}_{Y} \circ \mathcal{L}_{X}=\mathcal{L}_{[X, Y]}$, valid for $k$-forms. For vector fields is nothing but Jacobi identity.
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Author of the notes: Antonio J. Pan-Collantes
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