An electron is known to be in an orbital with $l = 2$. You can use the eigenstate of $L_z$, the z-component of orbital angular momentum, as basis of this, $l = 2$ subspace and denote them $|2, m_l\rangle$.
(a) Find the matrix representation of the operators $L_x$, $L_y$, and $L_z$ in this basis.
(b) Verify explicitly that $[L_x, L_y] = i\hbar L_z$ in the $l = 2$ subspace.
(c) What are possible values of the total angular momentum number $j$.
(d) If the electron is in a state with lowest $j$ (among those which you found in part (c)), what are the possible results of a measurement of $J_z$, the z-component of orbital angular momentum ?
(e) Suppose that the measurement of $J_z$ in part (d) resulted in $m_j = j$. If you now measure $L_z$, the z-component of orbital angular momentum, what are possible outcomes?