= Solution
Let $f$ and $g$ be <conserved quantities>, allowing explicit time dependence. Then
$$
\partial_t f=-\{f,H\},
\qquad
\partial_t g=-\{g,H\}.
$$
Using bilinearity of the <Poisson bracket>, commutation of partial differentiation with it, and the <Jacobi identity for the Poisson bracket>,
$$
\begin{aligned}
\frac d{dt}\{f,g\}
&=\{\{f,g\},H\}+\{\partial_t f,g\}+\{f,\partial_t g\}\\
&=\{\{f,g\},H\}-\{\{f,H\},g\}-\{f,\{g,H\}\}=0.
\end{aligned}
$$
\b[Thus $\{f,g\}$ is also conserved.]
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