.

(* restored equation*)

= yp_z zp_x -zp_x yp_z + z p_y xp_z - xp_z z p_y (33)

= y p_x (p_z z -zp_z ) + x p_y (z p_z - p_z z) (34)

$$= y p_x (-\imath \hbar) + x p_y (\imath\hbar)$$ (35)

$$= \imath \hbar ( x p_y -y p_x ) = \imath \hbar L_z = [L_x,L_y]$$ (36)

that

$$[L_x,L_y]= \imath \hbar L_z$$ (37)

$$[L_y,L_z]= \imath \hbar L_x$$ (38)

$$[L_z,L_x]= \imath \hbar L_y$$ (39)

we derive that

$$[L_x,L^+]= L_x (L_x+\imath L_y) - (L_x+\imath L_y) L_x$$

which is, expanding:

$$= L_x L_x+ \imath L_x L_y - L_x L_x- \imath L_y L_x = L_x L_x... ...L_x L_y - L_y L_x) = + \imath (\imath \hbar L_z) = - \hbar L_K$$