On page 203 3rd edition of Schiff we are given the angular momentum matrices ${J}$ for $j=1$.
I am curious as to how these relate to orbital angular momentum for $j = 1$. If we take the corresponding 3x3 matrices for spin given on page 198. Lets just use
$$S_x = i \hbar \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\ \end{pmatrix}\,\,\,\mathrm{and}\,\,\,S_y = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \\ \end{pmatrix}$$
With
$$J_x = \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{pmatrix}\,\,\,\mathrm{and}\,\,\,J_y = \frac{\hbar}{\sqrt{2}}\begin{pmatrix} 0 & -i & 1 \\ i & 0 & -i \\ 0 & i & 0 \\ \end{pmatrix}$$
Using $\vec{J} = \vec{L} + \vec{S}$ we can solve for $L_x$ and $L_y$ by subtracting $J$ from $S$.
$$L_x = J_x-S_x = \begin{pmatrix} 0 & \frac{\hbar}{\sqrt{2}} & 0 \\ \frac{\hbar}{\sqrt{2}} & 0 & i\hbar + \frac{\hbar}{\sqrt{2}} \\ 0 & -i\hbar + \frac{\hbar}{\sqrt{2}} & 0 \\ \end{pmatrix} $$ $$ L_y = J_y-S_y = \begin{pmatrix} 0 & -\frac{i\hbar}{\sqrt{2}} & -i \hbar \\ \frac{i \hbar}{\sqrt{2}} & 0 & -\frac{i\hbar}{2} \\ i \hbar & \frac{i \hbar}{\sqrt{2}} & 0 \\ \end{pmatrix}$$
$$L_z = J_z - S_z = \begin{pmatrix} \hbar & i \hbar& 0\\ -i \hbar & 0 & 0 \\ 0 & 0 & -\hbar \\ \end{pmatrix}$$ My question is that when I do this, they don't obey the standard commutation relations $[L_i,L_j] = i \hbar \epsilon_{ijk} L_k$. More curious is the 2x2 case for $j=1/2$ on page 203 again, you get null orbital angular momentum matrices. What went wrong here and what am I missing?