[QUE/CM-04005]

\( \newcommand{\PB}[1]{ [#1]_\text{PB} } \)

Without using the definition of the Poisson bracket, using only the properties given a single particle show that the components of angular momentum satisfy the Poisson bracket relations  $$\PB{L_1,L_2} = L_3, \qquad \PB{L_2,L_3}=L_1, \qquad \PB{L_3,L_1}=L_2$$  and $$ \PB{\vec{L}^2, L_k}=0, \qquad k=1,\cdots, n $$