Write Sz in the J angular momentum basis?

[dipole]

I'm working on a problem where I want to write the operator $S_z$ down in terms of some operator(s) in the $\vec{J} = \vec{L} + \vec{S}$ basis so that I can operate $S_z$ on the states $\mid \ell, s=1/2, j= \ell\pm1/2, m\rangle$ but I'm having trouble finding the correct combination of operators to do so.

Is this possible, and if so could anyone point me in the right direction? Thanks.

 

[Lavabug]

If you're using z as the preferred orientation for your problem(Jz), and your orbital and spin AM are in the same direction, you are already working in a basis with elements that are tensor products $\|j,m_j\rangle_z \otimes |s,m_s \rangle_z$, which are still eigenvectors of Sz with the same eigenvalues. Sz in this "wider" basis constructed from Lz and Sz basis vectors simply acts like the unit matrix on the j-part.