Addition of angular momenta

Please help with the following problems.

Suppose you know that an electron is in l = 1 state. As usual we refer the total angular momentum by J = L + S.
(a) What does | j = 3/2, j2 = 3/2> state correspond to in terms of the states in the product basis, | l2, s2 >?
(b) By acting J_ on this state find | j = 3/2, j2 = 1/2 >
(c) In the product basis, what are the states with j2 = 1/2?
(d) You already found the combination of the part (c) states which correspond to j = 3/2. Now, find the state with j = 1/2, using the fact that it must be orthogonal to the j = 3/2 state.

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...=1,lz = 1>

L- only acts on the ket with the orbital angular momentum while S- only acts on the ket with the spin:

L-|s=1/2,sz = 1/2>|l=1,lz = 1> = |s=1/2,sz = 1/2>L-|l=1,lz = 1> =sqrt(2) |s=1/2,sz = 1/2>|l=1,lz = 0>

S-|s=1/2,sz = 1/2>|l=1,lz = 1> = |s=1/2,sz = -1/2>|l=1,lz = 1>

So, we have:

|j = 3/2, jz = 1/2> = sqrt(2/3) |s=1/2,sz = 1/2>|l=1,lz = 0> + 1/sqrt(3) |s=1/2,sz ...