the set of integer pairs $\{(r,m)\mid r={\ell},{\ell}+1,\ m=-j,\dots,j\}$. Consider the lexicographic ordering $\varrho:\{1,\dots,4{\ell}+4\}\to \mathcal{J}_{\ell}$. For $j,k=1,\dots, 4{\ell}+4$, let $e_{j,k}$ be the $(4{\ell}+4)$-square matrix whose entries are all zero, but the one at line $j$ and column $k$ which is equal to $1$. Define $$E_{j,k} = e_{j,k} - e_{k,j}, \  F_{j,k} = i e_{j,k} + i e_{k,j}, \ D_{j,k} = ie_{j,j} - i e_{k,k} .$$

By a slight abuse of language, also set $e_{\varrho(j),\varrho(k)}=e_{j,k}$. The analogous identification can be used to define $E_{\varrho(j),\varrho(k)},F_{\varrho(j),\varrho(k)},D_{\varrho(j),\varrho(k)}$. Note that $$\label{Ji}
J_{i}(E_{({\ell},m),({\ell}+1,n)}) =- F_{({\ell},m),({\ell}+1,n)},
\quad
\mbox{ 
and
} \quad
J_{i}(F_{({\ell},m),({\ell}+1,n)}) = E_{({\ell},m),({\ell}+1,n)}.$$ Thanks to this notation we can conveniently represent the matrices corresponding to the controlled vector field (projected on $\mathcal{H}_{\ell}$). A computation shows that the control potentials in the $x$ and $y$ directions, $-i \cos \varphi\sin \theta$ and $-i \sin \varphi\sin \theta$ respectively, projected on $\mathcal{H}_{\ell}$, have the matrix representations $$\begin{aligned}
B_{1}^{(4{\ell}+4)} &= \sum_{m=-{\ell}}^{{\ell}}( - q_{{\ell},m} F_{({\ell},m),({\ell}+1,m-1)} + q_{{\ell},-m} F_{({\ell},m),({\ell}+1,m+1)})\\
B_{2}^{(4{\ell}+4)} &= \sum_{m=-{\ell}}^{{\ell}}(  q_{{\ell},m} E_{({\ell},m),({\ell}+1,m-1)} +q_{{\ell},-m} E_{({\ell},m),({\ell}+1,m+1)}),\end{aligned}$$ where $$q_{{\ell},m}= \sqrt{\frac{({\ell}-m+2)({\ell}-m+1)}{4(2{\ell}+1)(2{\ell}+3)}}.$$

Similarly, we associate with the control potential in the $z$ direction, $-i \cos \theta$ the matrix representation $$B_{3}^{(4{\ell}+4)}= \sum_{m=-{\ell}}^{{\ell}} p_{{\ell},m} F_{({\ell},m),({\ell}+1,m)},$$ with $$p_{{\ell},m}=- \sqrt{\frac{({\ell}+1)^{2} - m^{2}}{(2{\ell}+1)(2{\ell}+3)}}.$$

Useful bracket relations  {#useful}
-------------------------

From the identity $$[e_{j,k},e_{n,m}] 