 Solution

For $L_i={ϵ}_{ijk}{q}_j{p}_k$, the canonical Poisson brackets give $\{L_i,L_j\}={ϵ}_{ijk}{L}_k$. Consequently projections onto fixed axes $n,m$ obeyIf the two original projections are conserved and the axes are linearly independent, the preceding Jacobi identity argument makes the projection along $n\times m$ conserved as well. The vectors $n,m,n\times m$ form a basis of three-dimensional space, so every fixed-axis projection is a linear combination of these conserved quantities. Angular momentum is therefore conserved along every axis.