 Solution

$L_a={\epsilon }_{abc}{x}_b{p}_c$, with $[x_a,p_b]=i\hslash {\delta }_{ab}$ and position-position and momentum-momentum commutators zero. Expansion gives $[L_a,L_b]=i\hslash {\epsilon }_{abc}{L}_c$ and $[L_a,p_d]=i\hslash {\epsilon }_{adc}{p}_c$. Thus $[L_1,p_2]=i\hslash p_3\ne 0$, so they are not simultaneously diagonalisable.