.

(*restored equation*)

$$= L_z L_x + \imath L_z L_y - L_x L_z - \imath L_y L_z$$

which is

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

or

$$=\imath \hbar( L_y + (-\imath L_x ))= \imath^2\hbar (\imath L_y - \imath^2 L_x)$$

$$= -\hbar (\imath L_y + L_x) = -\hbar L^+$$

so that, we finally obtain

$$[L_z,L^+]= -\hbar L^+$$ (42)

and its companion

$$[L_z,L^-]= +\hbar L^-$$ (43)

Equations 42 and 43 tell us that L⁺ ladders us up, and L^- ladders us down on L_z components. If

L_z |> = something|>

i.e., from Equation 22 $L_z = -\imath \hbar \frac{\partial}{\partial \phi}$we have (already know that)

$$L_z \vert m_\ell> = m_K \hbar\vert m_\ell>$$