= Solution
Use the <commutator derivation identity> $[A,BC]=[A,B]C+B[A,C]$ and the <orbital angular momentum commutation relations>. They give
$$
\begin{aligned}
[\hat j_z,\hat j_x^2]&=i\hbar(\hat j_y\hat j_x+\hat j_x\hat j_y),\\
[\hat j_z,\hat j_y^2]&=-i\hbar(\hat j_x\hat j_y+\hat j_y\hat j_x),\\
[\hat j_z,\hat j_z^2]&=0.
\end{aligned}
$$
The first two expressions cancel on addition, so
$$
\boxed{[\hat j_z,\hat{\mathbf j}^{\,2}]=0.}
$$
The same argument works for each component of the <angular momentum operator>.
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