If $R(\alpha,\beta,\gamma)$ is the Rotation operator and $\alpha,\beta,\gamma$ are Euler angles and $J$ is the total angular momentum then how to get to this: $$[J^2,R]~=~0?$$
This is stated in Zettili's book Quantum Mechanics.
The rotation operator $R$ is generated by the angular momentum operators $J_i$: $R$ can be written as a power series of $J_i$.
Since $[J^2,J_i]=0$ for $i=1,2,3$ the operator $J^2$ will commute with every term in the power series of $R$. By this we have $[R,J^2]=0$.