Solution
= Solution
Hermiticity of the Cartesian <angular momentum> components gives
$$
\boxed{L_+^\dagger=L_-,\qquad L_-^\dagger=L_+.}
$$
The <angular momentum commutation relations> $[L_i,L_j]=i\hbar\epsilon_{ijk}L_k$ imply
$$
\boxed{[L^2,L_\pm]=0,\qquad [L_z,L_\pm]=\pm\hbar L_\pm.}
$$
For example, $[L_z,L_x+iL_y]=i\hbar L_y+\hbar L_x=\hbar L_+$. The two terms in $[L_x^2+L_y^2+L_z^2,L_i]$ cancel in pairs, giving the first <commutator>. Also $[L_+,L_-]=2\hbar L_z$.