Definition of Classical Possion Bracket:

$$[f, g]=\sum_{i=1, 2,\ldots, n}\frac{\pp(f, g)}{\pp(p_i, q_i)}=\sum_{i=1, 2,\ldots, n}\left(\frac{\pp f}{\pp p_i}\frac{\pp g}{\pp q_i}-\frac{\pp f}{\pp q_i}\frac{\pp g}{\pp p_i}\right)$$

If we define $$p_{-i}\mdef q_{i}$$, and $$I=\{\pm 1, \pm, 2,\ldots, \pm n\}$$, we can rewrite it:

$$[f, g]=\sum_{i\in I} \sgn(i) \frac{\pp f}{\pp p_i}\frac{\pp g}{\pp p_{-i}}$$

Jacobi Identity is $$[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0$$. With the new notation, we can rewrite it as

\begin{aligned} \sum_{\overrightarrow{fgh}}[f,[g,h]] &=\sum_{\overrightarrow{fgh}}\sum_{i\in I}\sgn(i)\frac{\pp f}{\pp p_i}\frac{\pp}{\pp p_{-i}}\left(\sum_{j\in I}\sgn(j)\frac{\pp g}{\pp p_j}\frac{\pp h}{\pp p_{-j}}\right)\\ &=\sum_{\overrightarrow{fgh}}\sum_{i, j\in I}\sgn(ij)\left(\frac{\pp f}{{\pp p_{i}}}\frac{\pp h}{{\pp p_{-j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{j}}} +\frac{\pp f}{{\pp p_{i}}}\frac{\pp g}{{\pp p_{j}}}\frac{\pp^2h}{{\pp p_{-i}}{\pp p_{-j}}}\right)\\ &=\sum_{\overrightarrow{fgh}}\sum_{i, j\in I}\sgn(ij)\left(\frac{\pp f}{{\pp p_{i}}}\frac{\pp h}{{\pp p_{-j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{j}}} +\frac{\pp h}{{\pp p_{i}}}\frac{\pp f}{{\pp p_{j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{-j}}}\right)\\ &=\sum_{\overrightarrow{fgh}}\sum_{i, j\in I}\sgn(ij)\left(\frac{\pp f}{{\pp p_{i}}}\frac{\pp h}{{\pp p_{-j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{j}}} +\frac{\pp f}{{\pp p_{i}}}\frac{\pp h}{{\pp p_{j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{-j}}}\right)\\ &=\sum_{\overrightarrow{fgh}}\sum_{i, j\in I}\sgn(ij)\left(\frac{\pp f}{{\pp p_{i}}}\frac{\pp h}{{\pp p_{-j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{j}}} -\frac{\pp f}{{\pp p_{i}}}\frac{\pp h}{{\pp p_{-j}}}\frac{\pp^2g}{{\pp p_{-i}}{\pp p_{j}}}\right)\\ &=0\end{aligned}