Let $X^H$ and $X^F$ be the Hamiltonian vector fields for the functions $H$ and $F$, i.e. $$ X^H\colon=\sum_j\pa_{p_j}H\pa_{q_j}-\pa_{q_j}H\pa_{p_j} $$ Verify that $[X^H,X^F]$ is the Hamiltonian vector fields for the function $-\{H,F\}$.

For simplicity put $n=1$, then $$ X^H=\pa_{p}H\pa_{q}-\pa_{q}H\pa_{p}, X^F=\pa_{p}F\pa_{q}-\pa_{q}F\pa_{p}, \{H,F\}=\pa_qH\pa_pF-\pa_pH\pa_qF $$ and thus: \begin{eqnarray*} X^{\{H,F\}} &=&(\pa_p(\pa_qH\pa_pF-\pa_pH\pa_qF))\pa_q -(\pa_q(\pa_qH\pa_pF-\pa_pH\pa_qF))\pa_p\\ &=&(\pa_qH\pa_p^2F+\pa_p\pa_qH\pa_pF-\pa_pH\pa_q\pa_pF-\pa_p^2H\pa_qF)\pa_q\\ &&-(\pa_q^2H\pa_pF+\pa_qH\pa_q\pa_pF-\pa_q\pa_pH\pa_qF-\pa_pH\pa_q^2F)\pa_p \end{eqnarray*} On the other hand we get for the commutator \begin{eqnarray*} [X^H,X^F] &=&(\pa_pH\pa_q\pa_pF-\pa_qH\pa_p\pa_pF)\pa_q +(-\pa_pH\pa_q\pa_qF+\pa_qH\pa_p\pa_qF)\pa_p\\ &&+(-\pa_pF\pa_q\pa_pH+\pa_qF\pa_p\pa_pH)\pa_q +(\pa_pF\pa_q\pa_qH-\pa_qF\pa_p\pa_qH)\pa_p\\ &=&(\pa_pH\pa_q\pa_pF-\pa_qH\pa_p^2F-\pa_q\pa_pH\pa_pF+\pa_p^2H\pa_qF)\pa_q\\ &&+(-\pa_pH\pa_q^2F+\pa_qH\pa_p\pa_qF+\pa_q^2H\pa_pF-\pa_p\pa_qH\pa_qF)\pa_p \end{eqnarray*}