= Solution
For $L_i=\epsilon_{ijk}q_jp_k$, the canonical <Poisson brackets> give $\{L_i,L_j\}=\epsilon_{ijk}L_k$. Consequently projections onto fixed axes $n,m$ obey
$$
\{n\cdot L,m\cdot L\}=(n\times m)\cdot L.
$$
If 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. \b[Angular momentum is therefore conserved along every axis.]
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