Given a system of two spin $\frac{1}{2}$ particles, I need to find the matrix representation for
$$ \hat{J_i} = \hat{L_i}\otimes \hat{I}+\hat{I}\otimes \hat{S_i }$$ for $i=1,2,3,+,-$ in both, the coupled $|j,m_j,l,s\rangle$ and uncoupled $|m_l,m_s,l,s\rangle$ basis. Then, the change of basis between them. For this one I have found the following relations between the basis kets:
| $|j,m_j,l,s\rangle$ | $|m_l,m_s,l,s\rangle$ |
|---|---|
| $\big|1,1,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\big|\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle$ |
| $\big|1,0,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\frac{1}{\sqrt{2}}(|\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle + \big|-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle)$ |
| $|1,-1,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\big|-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle$ |
| $\big|0,0,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\frac{1}{\sqrt{2}}(\big|\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle -\big|-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle)$ |
This allows deducing the matrix for change of basis from $|j,m_j,l,s\rangle$ to $|m_l,m_s,l,s\rangle$:
$$ P_{jm->ls} \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}}\\ 0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}\\ 0 & 0 & 1 & 0\\ \end{pmatrix} $$
Where the order chosen for each base was:
| $|j,m_j,l,s\rangle$ | $|m_l,m_s,l,s\rangle$ |
|---|---|
| $\big|1,1,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\big|\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle$ |
| $\big|1,0,\frac{1}{2},\frac{1}{2}\big\rangle$ | $|\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle$ |
| $|1,-1,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\big|-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle$ |
| $\big|0,0,\frac{1}{2},\frac{1}{2}\big\rangle$ | $\big|-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\big\rangle$ |
For instance, the matrix representation for $J_1$ that I have found in the basis $|j,m_j,l,s\rangle$, is $$ J_1^j = \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}. $$
And for the basis $|m_l,m_s,l,s\rangle$ $$ J_1^{ls} = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0\\ \end{pmatrix}. $$ For the last one, I first found the representations $L_1$ and $S_1$, in the basis $|m_l,m_s,l,s\rangle$ with the definition of the ladder operators $L_1 = \frac{L_{+} + L_{-}}{2}$ and $L_2 = \frac{L_{+} - L_{-}}{2i}$, and then $J_1^{ls} = L_1+S_1$.
The problem I have is in obtaining one matrix from the other through the change of base matrix. I claim that $J_1^{ls} = P_{jm->ls}J_1^{j}$. But, it is not true from what I have obtained. There is something I'm missing? thanks in advance.
Answer: After checking the links below, it turns out that the way these matrices are related is $J_1^{ls} = (P_{jm->ls})J_1^{j}(P_{jm->ls})^{T}$.
