# How to decompose Cartesian into spherical tensor with Wigner-Eckart theorem?

I'm solving a question that required me to work with a Cartesian tensor, $$A_i$$, with Wigner-Eckart theorem.

The book was Sakurai Modern Quantum Mechanics with information given in related posts:

The issue I have had was that the form of Wigner-Eckart theorem was given in spherical tensor. But $$A_i$$ was a cartesian tensor.

Qa. How to decompose $$A_i$$ into $$T_{q}^{(k)} = Y_{l=k}^{m=q}(A_i)$$? Is it a summation over index $$m/q,l/k$$? Does it have coefficients? And how to determine the rank of the tensor?

There was a projection theorem when $$j=j'$$, Eq 3.11.40 $$\langle \alpha', jm'|V_q |\alpha, jm\rangle = \frac{\langle \alpha', jm'|J\cdot V|\alpha, jm\rangle} {\hbar^2 j(j+1)} \langle jm' |J_q|jm\rangle$$

However, it was very confusing. From the reading, the book seemed to indicate that the subscript $$V_q$$ here indicate the spherical $$m=q$$.

Qb. Could projection theorem be used to deal with Cartesian tensor?