Formulas for Lie derivative, exterior derivative, brackets, interior and wedge products

For the definitions see:
Lie derivative of forms,
Lie derivative for tensors in general,
interior product,
Lie bracket,
exterior derivative,...

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. ${\mathcal{L}}_{fX}\omega =f{\mathcal{L}}_X\omega +df\wedge i_X\omega$ (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. $i_{[X,Y]}=[{\mathcal{L}}_X,i_Y]$ (warning: informal notation). It is the same as equation (1.62) in @olver86: $\mathbf{v}(\mathbf{w}⌟\omega )=[\mathbf{v},\mathbf{w}]⌟\omega +\mathbf{w}⌟\mathbf{v}(\omega )$. 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. ${\mathcal{L}}_X(⟨\omega ;Y⟩)=⟨{\mathcal{L}}_X\omega ;Y⟩+⟨\omega ;{\mathcal{L}}_{X}Y⟩$ or, in general for a $k$-form,
   3. ${\mathcal{L}}_X(⟨\omega ;Y_1,\dots ,Y_k⟩)=⟨{\mathcal{L}}_X\omega ;Y_1,\dots ,Y_k⟩+⟨\omega ;{\mathcal{L}}_X{Y}_1,\dots ,Y_k⟩+\dots$. See @olver86 page 74 exercise 1.35.
5. infinitesimal Stokes' theorem
6. Cartan formula
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.