The Poisson bracket possesses the symplectic structure
(Note that this algebraic structure is not commutative)
Apply this rule to the brackets:
Assemble the three brackets to prove that
Following this result, we apply the same rules to the following brackets:
We note that the order of is significant for the bracket operator. Adding the three brackets will yield 0.
Check out my other notes at Proof, Disproof, and Derivation