A basic question on representation theory

[nickthequick]

Homework Statement

Hey, this is problem 4A out of Georgi (Lie Algebras in particle physics). The question says given an operator $$O_x$$ , x=1,2, in the spin 1/2 rep of SU(2). $$[J_a,O_x]=O_y(\sigma_a)_{yx}/2$$ where $$\sigma_a$$ are the Pauli spin matrices.

Given
$$A=\langle 3/2, -1/2, \alpha|O_1|1,-1,\beta\rangle$$
find

$$B=\langle 3/2, -3/2, \alpha|O_2|1,-1,\beta\rangle$$

Homework Equations

This seems to be a basic application of the Wigner Eckart theorem, I just cannot seem to make any fruitful progress. Any comments/ hints would be appreciated.

 

[BigJDubs]

The Attempt at a Solution I guess the first step is to find \langle 3/2, -1/2|O_1|1,-1\rangle. We know that [J_a,O_x]=O_y(\sigma_a)_{yx}/2 , so we can say \langle 3/2, -1/2|[J_z,O_1]|1,-1\rangle = \langle 3/2, -1/2|O_2|1,-1\rangle (\sigma_z)_{21}/2. I am stuck here.