Showing the interior derivative of a differential form satisfies certain properties I'm quite new to differential forms/interior derivatives so if anyone could help with understanding the intuition behind proving this certain property step by step would be great.
Let $\iota_X$ denote the interior derivative of differential forms with a vector field $X$. Show that the operator: $$L_X:= \iota_X \ \circ d \ + \ d \ \circ \ \iota_X$$
satisfies $$L_X(\omega_1 \ \wedge \ \omega_2)=(L_X \omega_1) \ \wedge \omega_2 \ + \ \omega_1 \ \wedge \ (L_X \omega_2)$$
So far I have gotten to the expression:
$$\iota_X \ \circ \ [(d\omega_1) \ \wedge \ \omega_2 \ + \ (-1)^k\omega_1 \ \wedge \ d(\omega_2)] \ + d \ \circ \ [(\iota_X\omega_1) \ \wedge \ \omega_2 \ + \ (-1)^k\omega_1 \ \wedge \iota_X(\omega_2)]$$
by using the following properties:
$$\text{(1)} \ \ \ d(\omega_1 \wedge \omega_2)=d\omega_1 \wedge \omega_2 +(-1)^k(\omega_1 \wedge d\omega_2)$$
$$\text{(2)} \ \ \ \iota_X(\omega_1 \wedge \omega_2) = \iota_X(\omega_1) \wedge \omega_2 + (-1)^k\omega_1 \wedge \iota_X(\omega_2)$$
Edit: after substituting the properties again for a second time I am left with:
$$\iota_x \ \circ \ d\omega_1 \ \wedge \ \omega_2 \ + \ (-1)^k d\omega_1 \ \wedge \ \iota_x(\omega_2)  \ + \ (-1)^k \iota_x \omega_1 \ \wedge d\omega_2 \ + \ (-1)^{2k} \omega_1 \ \wedge \ (\iota_X \ \circ \ d\omega_2) \ + \ d \ \circ \ \iota_X \omega_1 \ \wedge \ \omega_2 \ + \ (-1)^k(\iota_X \omega_1 \ \wedge \ d\omega_2) \ + \ (-1)^k(d\omega_1) \ \wedge \ \iota_X \omega_2 \ + \ (-1)^{2k}\omega_1 \ \wedge \ (d \ \circ \ \iota_x\omega_2)$$
 A: Following from above:
\begin{align} 
L_X(\omega_1 \wedge \omega_2) &= (\iota_X \circ d + d \circ \iota_X)(\omega_1 \wedge \omega_2) = (\iota_X \circ d)(\omega_1 \wedge \omega_2)+ (d \circ \iota_X)(\omega_1 \wedge \omega_2)\\ 
&= \iota_X \circ [(d\omega_1) \wedge \omega_2 + (-1)^k\omega_1 \wedge d(\omega_2)]+ d\circ [(\iota_X \omega_1) \wedge \omega_2+(-1)^k\omega_1 \wedge\iota_X(\omega_2)]\\ 
&= \iota_X \circ d\omega_1  \wedge \omega_2 + (-1)^{k+1} d\omega_1 \wedge \iota_X\omega_2 + (-1)^k \iota_X \omega_1 \wedge d\omega_2 + (-1)^{2k} \omega_1 \wedge \iota_X \circ d\omega_2 \\ & \qquad + d\circ\iota_X \omega_1 \wedge \omega_2  +  (-1)^{k-1}\iota_X \omega_1 \wedge d\omega_2 + (-1)^k d\omega_1 \wedge \iota_X \omega_2 + (-1)^{2k}\omega_1 \wedge d\circ\iota_X\omega_2\\ 
&= (\iota_X \circ d\omega_1 \wedge \omega_2  + d\circ \iota_X\omega_1 \wedge \omega_2) + (\omega_1 \wedge \iota_X \circ d\omega_2 + \omega_1 \wedge d\circ \iota_X \omega_2)\\
&= (L_X\omega_1) \wedge \omega_2 + \omega_1 \wedge (L_X \omega_2)\\ 
\end{align}
Thank you for the help Ted!:)
