Let $SO(3)$ be generated by infinitesimal rotations $L_x, L_y, L_z$ such the typical relations $ [L_x, L_y] = L_z $ and similar. Let $L_\pm = L_x \pm i L_y$ be the raising and lowering operators. Is there a close formula for $[L_+^m, L_z^n]$ ?
I tried finding some kind of Leibniz rule. I came up with $[A,BC]=[A,B]C + B[A,C]$ and this could be extended to several variables. The result:
$$ [L_+^m, L_z^n] = [L_+^m, L_z]L_z^{n-1} + L_z^1[L_+^m, L_z]L_z^{n-2} + \dots + L_z^{n-1}[L_+^m, L_z]$$
This may be more tractable than I thought if I can identify a formula for: $[L_+^m, L_z]$.