Any spherically symmetric function of the canonical coordinate and momentum of a particle can depend only on \(r^{2}, p^{2}\), and \(\mathbf{r} \cdot \mathbf{p}\). Show that the Poisson bracket of any such function \(f\) with a component of the particle's angular momentum is zero. In particular, show that \(\left\{L_{z}, f\right\}=0\), where \(L_{z}=(\mathbf{r} \times \mathbf{p})_{z}\).