Theorem: Poisson Bracket satisfies Jacobi Identity/Proof 1

For $n \in \N$, let:
:$\map f {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
:$\map g {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
:$\map h {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
be real functions, dependent on time and the canonical variables $\tuple {\mathbf q, \mathbf p}$.
Let $\sqbrk {f, g}$ denote the '''Poisson bracket''' of $f$ and $g$.
Then:
:$\sqbrk {f, \sqbrk {g, h} } + \sqbrk {g, \sqbrk {h, f} } + \sqbrk {h, \sqbrk {f, h} } = 0$
That is, the Poisson bracket operation satisfies the Jacobi identity.

Proof:
The Poisson bracket of two functions $f$ and $g$ on a phase space with coordinates $\tuple {\mathbf q, \mathbf p, t} = \tuple {q_1, \dots, q_n, p_1, \dots, p_n, t}$ is defined as:
:$\ds \sqbrk {f, g} = \sum_i \paren {\frac {\partial f} {\partial q_i} \frac {\partial g} {\partial p_i} - \frac {\partial f} {\partial p_i} \frac {\partial g} {\partial q_i} }$
Likewise, the brackets $\sqbrk {g, h}$ and $\sqbrk {h, f}$ are:
:$\ds \sqbrk {g, h} = \sum_i \paren {\frac {\partial g} {\partial q_i} \frac {\partial h} {\partial p_i} - \frac {\partial g} {\partial p_i} \frac {\partial h} {\partial q_i} }$
:$\ds \sqbrk {h, f} = \sum_i \paren {\frac {\partial h} {\partial q_i} \frac {\partial f} {\partial p_i} - \frac {\partial h} {\partial p_i} \frac {\partial f} {\partial q_i} }$
We have that the Poisson bracket, $\sqbrk {f, \sqbrk {g, h} }$ is defined as:
:$\ds \sqbrk {f, \sqbrk {g, h} } = \sum_i \paren {\frac {\partial f} {\partial q_i} \frac {\partial} {\partial p_i} \sqbrk {g, h} - \frac {\partial f} {\partial p_i} \frac {\partial} {\partial q_i} \sqbrk {g, h} }$
Expanding the expression for $\sqbrk {g, h}$ gives:
{{begin-eqn}}
{{eqn | l = \sqbrk {f, \sqbrk {g, h} }
      | r = \sum_i \paren{\frac{\partial f}{\partial q_i} \frac{\partial}{\partial p_i} \paren{\sum_j \paren{\frac{\partial g}{\partial q_j} \frac{\partial h}{\partial p_j} - \frac{\partial g}{\partial p_j} \frac{\partial h}{\partial q_j} } } - \frac{\partial f}{\partial p_i} \frac{\partial}{\partial q_i} \paren{\sum_j \paren{\frac{\partial g}{\partial q_j} \frac{\partial h}{\partial p_j} - \frac{\partial g}{\partial p_j} \frac{\partial h}{\partial q_j} } } }
}}
{{eqn | r = \sum_i\sum_j \paren{ \frac{\partial f}{\partial q_i} \paren{ \frac{\partial}{\partial p_i} \paren{\frac{\partial g}{\partial q_j}\frac{\partial h}{\partial p_j} } - \frac{\partial}{\partial p_i} \paren{\frac{\partial g}{\partial p_j}\frac{\partial h}{\partial q_j} } } - \frac{\partial f}{\partial p_i} \paren{ \frac{\partial}{\partial q_i} \paren{\frac{\partial g}{\partial q_j}\frac{\partial h}{\partial p_j} } - \frac{\partial}{\partial q_i} \paren{\frac{\partial g}{\partial p_j}\frac{\partial h}{\partial q_j} } } }
}}
{{eqn | r =  \sum_i\sum_j\paren{\frac{\partial f}{\partial q_i}\paren{\paren{\frac{\partial ^2 g}{\partial p_i \partial q_j}\frac{\partial h}{\partial p_j} + \frac{\partial g}{\partial q_j}\frac{\partial ^2 h}{\partial p_i \partial p_j} } - \paren{\frac{\partial ^2 g}{\partial p_i \partial p_j}\frac{\partial h}{\partial q_j} + \frac{\partial g}{\partial p_j}\frac{\partial ^2 h}{\partial p_i \partial q_j} } } - \frac{\partial f}{\partial p_i}\paren{\paren{\frac{\partial ^2 g}{\partial q_i \partial q_j}\frac{\partial h}{\partial p_j} + \frac{\partial g}{\partial q_j}\frac{\partial ^2 h}{\partial q_i \partial p_j} } - \paren{\frac{\partial ^2 g}{\partial q_i \partial p_j}\frac{\partial h}{\partial q_j} + \frac{\partial g}{\partial p_j}\frac{\partial ^2 h}{\partial q_i \partial q_j} } } }
}}
{{eqn | r = \sum_i\sum_j\paren{\frac{\partial f}{\partial q_i}\frac{\partial ^2 g}{\partial p_i \partial q_j}\frac{\partial h}{\partial p_j} + \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial q_j}\frac{\partial ^2 h}{\partial p_i \partial p_j} - \frac{\partial f}{\partial q_i}\frac{\partial ^2 g}{\partial p_i \partial p_j}\frac{\partial h}{\partial q_j} - \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_j}\frac{\partial ^2 h}{\partial p_i \partial q_j} - \frac{\partial f}{\partial p_i}\frac{\partial ^2 g}{\partial q_i \partial q_j}\frac{\partial h}{\partial p_j} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_j}\frac{\partial ^2 h}{\partial q_i \partial p_j} + \frac{\partial f}{\partial p_i}\frac{\partial ^2 g}{\partial q_i \partial p_j}\frac{\partial h}{\partial q_j} + \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial p_j}\frac{\partial ^2 h}{\partial q_i \partial q_j} }
}}
{{end-eqn}}
Similarly:
{{begin-eqn}}
{{eqn | l = \sqbrk {g, \sqbrk {h, f} }
      | r = \sum_i\sum_j\paren{\frac{\partial g}{\partial q_i}\frac{\partial ^2 h}{\partial p_i \partial q_j}\frac{\partial f}{\partial p_j} + \frac{\partial g}{\partial q_i}\frac{\partial h}{\partial q_j}\frac{\partial ^2 f}{\partial p_i \partial p_j} - \frac{\partial g}{\partial q_i}\frac{\partial ^2 h}{\partial p_i \partial p_j}\frac{\partial f}{\partial q_j} - \frac{\partial g}{\partial q_i}\frac{\partial h}{\partial p_j}\frac{\partial ^2 f}{\partial p_i \partial q_j} - \frac{\partial g}{\partial p_i}\frac{\partial ^2 h}{\partial q_i \partial q_j}\frac{\partial f}{\partial p_j} - \frac{\partial g}{\partial p_i}\frac{\partial h}{\partial q_j}\frac{\partial ^2 f}{\partial q_i \partial p_j} + \frac{\partial g}{\partial p_i}\frac{\partial ^2 h}{\partial q_i \partial p_j}\frac{\partial f}{\partial q_j} + \frac{\partial g}{\partial p_i}\frac{\partial h}{\partial p_j}\frac{\partial ^2 f}{\partial q_i \partial q_j} }
}}
{{eqn | l = \sqbrk {h, \sqbrk {f, g} }
      | r = \sum_i\sum_j\paren{\frac{\partial h}{\partial q_i}\frac{\partial ^2 f}{\partial p_i \partial q_j}\frac{\partial g}{\partial p_j} + \frac{\partial h}{\partial q_i}\frac{\partial f}{\partial q_j}\frac{\partial ^2 g}{\partial p_i \partial p_j} - \frac{\partial h}{\partial q_i}\frac{\partial ^2 f}{\partial p_i \partial p_j}\frac{\partial g}{\partial q_j} - \frac{\partial h}{\partial q_i}\frac{\partial f}{\partial p_j}\frac{\partial ^2 g}{\partial p_i \partial q_j} - \frac{\partial h}{\partial p_i}\frac{\partial ^2 f}{\partial q_i \partial q_j}\frac{\partial g}{\partial p_j} - \frac{\partial h}{\partial p_i}\frac{\partial f}{\partial q_j}\frac{\partial ^2 g}{\partial q_i \partial p_j} + \frac{\partial h}{\partial p_i}\frac{\partial ^2 f}{\partial q_i \partial p_j}\frac{\partial g}{\partial q_j} + \frac{\partial h}{\partial p_i}\frac{\partial f}{\partial p_j}\frac{\partial ^2 g}{\partial q_i \partial q_j} }
}}
{{end-eqn}}
Summing all three terms $\sqbrk {f, \sqbrk {g, h} }$, $\sqbrk {g, \sqbrk {h, f} }$, and $\sqbrk {h, \sqbrk {f, g}}$, they cancel out:
:$\ds \sqbrk {f, \sqbrk {g, h} } + \sqbrk {g, \sqbrk {h, f} } + \sqbrk {h, \sqbrk {f, g} } = 0$
Thus, the Poisson bracket satisfies the Jacobi identity.
{{qed}}