$H_{SO}=f(r)\overrightarrow{S}\cdot\overrightarrow{L} \,\,\,$ $[H_{SO},L_z] \neq 0$

Question

Let $$H_{SO}=f(r)\overrightarrow{S}\cdot\overrightarrow{L}$$ I know that: $$[H_{SO},L_z] \neq 0$$ $$[H_{SO},S_z]\neq 0 $$ but I can not get or find the precise result.

Answer 1

The operators ${\bf L}$ and ${\bf S}$ commute,

$$ [S_i, L_j] = 0 \tag{1} $$

So that

$$ [H_{\rm SO}, S_i] = [f(r)\sum_k S_k L_k, S_i] = \sum_{k}f(r)L_k \color{blue}{[S_k,S_i]} = \sum_{j,k}f(r)L_k \color{blue}{i\hbar \epsilon_{kij}S_j} = i\hbar f(r)\sum_{j,k} \epsilon_{ijk}S_jL_k \tag{2} $$