Schouten-Nijenhuis bracket computations

Fix a system of local coordinates and consider two vector fields

$$X=\sum_i a_i \del_{x_i},\quad Y=\sum_i b_i\del_{x_i},\quad X,Y\in\gX(M),\quad x_1,\hdots, x_n\text{ coordinates}$$

$$[X, Y]=\sum_i a_i\left(\sum_j \pd{b_j}{x_i}\del_{x_j}\right)- \sum_i b_i\left(\sum_j\pd{a_j}{x_i}\del_{x_j}\right).$$

Let $\zeta_i=\del_{x_i}$ and consider it as an odd formal variable

$$\zeta_i\zeta_j=-\zeta_j\zeta_i\quad (\del_{x_i}\land\del_{x_j}=-\del_{x_j}\land\del_{x_i}).$$

Then

$$X:=\sum_i a_i \zeta_i,\quad Y:=\sum_i b_i\zeta_i$$

$$\begin{align*}[X, Y]&=\sum_i\left(\pd{X}{\zeta_i}\pd{Y}{x_i}- \pd{Y}{\zeta_i}\pd... ...\ &= \left(\sum_i \del_{\zeta_i}\land\del_{x_i}\right)(X\tensor Y). \end{align*}$$
Extend this idea to multivector fields

$$P\in\gX^p(M),\quad P=\sum_{i_1<\hdots<i_p}\del_{x_{i_1}}\wed... ...i_1<\hdots<i_p} P_{i_i\hdots i_p}\zeta_{i_1}\hdots\zeta_{i_p}.$$

Fix the following differentiation rule
$$\begin{align*} \del_{\zeta_{i_p}}(\zeta_{i_1}\hdots\zeta_{i_p}) &=\zeta_{i_1}\hd... ...(-1)^{p-k}\zeta_{i_1}\hdots\Hat{\zeta_{i_k}}\hdots \zeta_{i_{p-1}}. \end{align*}$$
Then we claim that

$$[P, Q]_{SN}=\sum_{i}\del_{\zeta_i}P\,\del_{x_i}Q - (-1)^{(p-1)(q-1)}\del_{\zeta_i}Q\,\del_{x_i}P.$$