Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows
For Hermitian vector space of complex dimension $n$. we have followings. \begin{align} &\phantom{1}^* \Pi_{p,q} = \Pi^{*}_{n-q, n-p} \\ & [ L,w] = [L,J] = [L^*, w] = [L^*, J]=0 \\ & [ L^*, L]= \sum_{p=0}^{2n} (n-p) \Pi_p \end{align} where $w = \sum (-1)^{dr+r} \Pi_r$, $**=w$, $* w * = id$. thus $ww = id$ And $L^* = w * L *$.
Briefly explain about symbols, $\Pi_r$ is a linear mapping $\wedge F \rightarrow \wedge^r F$ and $\Pi_{p,q} : \wedge F \rightarrow \wedge^{p,q} F$. $J = \sum i^{p-q} \Pi_{p,q}$, $L : \wedge^{p,q} F \rightarrow \wedge^{p+1, q+1} F$.
What i want to do is prove above statement using direct commutator. The textbook proves this statement in a slightly different way, what i want to do is direct computation $i.e$ \begin{align} [L,w] = Lw - wL =?0 \end{align}