Poisson's Identity: Solving ((φλ)χ)+((λχ)φ)+((χφ)λ)=0

[aggarwal]

please help me to solve this identity

((\phi\lambda)\chi)+((\lambda\chi)\phi)+((\chi\phi)\lambda)=0


where () = poisson bracket

\phi=\phi(t,q_{i},p_{i})
\chi=\chi(t,q_{i},p_{i})
\lambda=\lambda(t,q_{i},p_{i})
for i=1,2,...,n

 

[dextercioby]

This is a very lengthy proof. For the beginning what is \left [\phi,\lambda]_{PB} eual to ?