I have a question regarding the alleged Lie algebra structure for $End(E)$-valued differential forms. In Well's book Differential analysis on complex manifolds, Chapter III, the following definition of the "Lie bracket" of $\alpha\in\Omega^p(M,End(E))$ and $\beta\in \Omega^q(M,End(E))$: choose a frame $f$ for $E$ and consider the naturally induced frame in $End(E)\simeq E\otimes E^*$.
Identify $\alpha \equiv \alpha(f)$ meaning $\alpha(f)$ is a matrix of p-forms and $\beta(f)$ is a matrix of q-forms. Then one defined the value of $[\alpha,\beta](f)$ as the matrix formed by $$ [\alpha,\beta](f) := \alpha(f)\wedge \beta(f) - (-1)^{pq}\beta(f)\wedge \alpha(f). $$ Here $\alpha(f)\wedge\beta(f)$ is to be interpreted as the matrix of $(p+q)$-forms with ($i,j)$ element given by $\sum_k \alpha(f)^i_{~k} \wedge\beta(f)^k_{~j}$. Wells claims that this defines a Lie algebra structure in $$ \Omega^*(M,End(E)) = \sum_{m\geq 0} \Omega^m(M,End(E)). $$ However, I think this claim is false since this Lie bracket fails to be skew-symmetric. Certainly $$ [\beta,\alpha](f) = (-1)^{pq+1} [\alpha,\beta](f) \neq - [\alpha,\beta](f) $$ I understand that, however, this is the natural structure to be given on $\Omega^*(M,End(E))$ which showcases a particular relation to Lie algebra valued forms. I have also seen this bracket written differently as $[\alpha\wedge\beta]$ to indicate that one is "commuting" the endomorphisms as well.
This particular question enlightens some of my concerns but doesn't quite address the issue of being a proper Lie bracket.
Edit: corrected wrong exponent $(-1)^{p+q}\rightarrow (-1)^{pq}$ in definition.
I think you're right. Look up https://en.wikipedia.org/wiki/Graded_Lie_algebra#Graded_Lie_superalgebras and by your computation you have such an algebra