= Solution
Use the <canonical commutation relations> $[\widehat x_i,\widehat x_j]=[\widehat p_i,\widehat p_j]=0$ and $[\widehat x_i,\widehat p_j]=i\hbar\delta_{ij}I$. For compactness write products without hats in the calculation, on a common domain such as smooth compactly supported wavefunctions. Expanding the <commutator> gives
$$
[L_1,L_2]=[x_2p_3,x_3p_1]-[x_2p_3,x_1p_3]-[x_3p_2,x_3p_1]+[x_3p_2,x_1p_3].
$$
The middle two <commutators> vanish because all the relevant indices differ. The first is $-i\hbar x_2p_1$, since $[p_3,x_3]=-i\hbar I$; the last is $i\hbar x_1p_2$, since $[x_3,p_3]=i\hbar I$. Thus
$$
\boxed{[\widehat L_1,\widehat L_2]=i\hbar(\widehat x_1\widehat p_2-\widehat x_2\widehat p_1)=i\hbar\widehat L_3.}
$$
This is one of the <orbital angular momentum commutation relations>.
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