Solution
= Solution
The <Jacobi identity for the Poisson bracket> is
$$
\boxed{\{F,\{G,H\}\}+\{G,\{H,F\}\}
+\{H,\{F,G\}\}=0}.
$$
For quantities with no explicit time dependence,
$$
\dot F=\{F,H\},
\qquad
\dot G=\{G,H\}.
$$
If $F$ and $G$ are conserved, the first two terms of the Jacobi identity vanish. Hence
$$
\{H,\{F,G\}\}=0,
$$
or, by antisymmetry,
$$
\boxed{\{\{F,G\},H\}=0}.
$$
\b[Thus the <poisson bracket of conserved quantities> is itself conserved.]