zero elements in the $n$th row and $n$th column. In particular there exists $M \in \mathrm{Lie}\left( \{\mathcal{E}_{2k-1}(B^{(n)})\,:\,k=1,\ldots,n-1\} \right) $ such that $M+\mathcal{E}_{2n-1}(B^{(n)}) $ has only two nonzero elements in the positions $(n-1,n)$ and $(n,n-1)$. So $$\begin{aligned}
&\mathrm{Lie} (  \{\mathcal{E}_{2k-1}(B^{(n)})\mid k=1,\ldots,n\} )  \\
&\qquad \supset \mathrm{Lie} \left(  \{\mathcal{E}_{2k-1}(B^{(n)})\mid k=1,\ldots,n-1\}  \cup \{M+\mathcal{E}_{2n-1}(B^{(n)})\} \right)
 = \mathfrak{su}(n).\end{aligned}$$ Therefore, thanks to Theorems \[metalemma\] and \[metalemma-tracking\], system  is approximately simultaneously controllable and a stalker.

The 3D molecule {#rotational}
===============

Let us go back to the system presented in the introduction for the orientation of a linear molecule, that is, $$\label{mol!}
i \hbar \dot \psi = -\Delta \psi + (u_1 \cos \theta + u_2 \cos \varphi\sin \theta+ u_3 \sin \varphi\sin \theta)\psi,$$ where $\psi(t) \in {\mathcal{H}}=L^2({S}^2,{{\mathbb{C}}})$.

A basis of eigenvectors of the Laplace–Beltrami operator $\Delta$ is given by the spherical harmonics $Y^m_{\ell}(\theta, \varphi)$, which satisfy $$\Delta Y^m_{\ell}(\theta,\varphi) = -{\ell}({\ell}+1)Y^m_{\ell}(\theta, \varphi).$$ The spectrum of $A = i\Delta$ is $\{ -i {\ell}({\ell}+1)\mid {\ell}\in {\mathbb{N}}\}$. Each eigenvalue $-i {\ell}({\ell}+1)$ is of finite multiplicity $2{\ell}+1$. Therefore $A$ satisfies Assumptions $(\mathbb{A}1)$ and $(\mathbb{A}5)$. Using the notations of the preceding sections we set $B_{1}$, $B_{2}$, $B_{3}$ to be the multiplication operators by $-i \cos \varphi\sin \theta$, $-i\sin \varphi\sin \theta$, $-i \cos \theta$ respectively. Being 