I am currently trying to make Exercise 22 of these lecture notes. I am stuck on what I believe to be maybe something trivial, but I can't for the life of me figure out how to tackle this question. I'll add the relevant information below.
The Question
For this question, we first need to define the "wedge" product between $K \in \Omega^p(M, \mathrm{End}(E)$ and $\eta \in \Omega^q(M, E)$. We define $$ (K \wedge \eta)(X_1, \ldots, X_{p + q}) = \sum_{\sigma} \mathrm{sign}(\sigma) K(X_{\sigma(1)}, \ldots, X_{\sigma(p)})\left( \eta(X_{\sigma(p+1)}, \ldots, \eta(X_{\sigma(p+q)}) \right), $$ where the sum is over all $(p,q)$-shuffles $\sigma$, i.e. all permuations $\sigma$ with $\sigma(1) < \ldots < \sigma(p)$ and $\sigma(p + 1) < \ldots < \sigma(p + q)$.
Any such $K$ induces a linear map $$ \hat K \colon \Omega^\bullet(M,E) \to \Omega^{\bullet + q}(M,E): \eta \mapsto K \wedge \eta. $$
Exercise 22. Show that $\hat K$ is an endomorphism of the graded (left) $\Omega(M)$-module $\Omega(M,E)$, i.e. according to the graded sign rule: $$ \hat K(\omega \wedge \eta) = (-1)^{pq} \omega \wedge \hat K(\eta). $$
My strategies, my work and what I find hard
I have tried to write out both sides of this equation, but that ends up in a mess. See my work here. I then tried to take a step back and thought if there was a way to use other properties that were already established. I thought of this: \begin{align*} \hat K(\omega \wedge \eta) &= K \wedge (\omega \wedge \eta) \\ &= (K \wedge \omega) \wedge \eta \\ &= (-1)^{pq} (\omega \wedge K) \wedge \eta \\ &= (-1)^{pq} \omega \wedge (K \wedge \eta) \\ &= (-1)^{pq} (\omega) \wedge \hat K(\eta). \end{align*}
The only thing that is bothering me is the associativity of the "wedge products". Trying to prove that associativity, brings about the same mess as before. Is the right strategy or am I missing something?
Thanks in advance for the help!