$B_{1}$, $B_{2}$, $B_{3}$ bounded, conditions $(\mathbb{A}2)$, $(\mathbb{A}3)$, and $(\mathbb{A}4)$ hold. Hence $(\mathbb{A}')$ is satisfied. Moreover, as proved in [@weakly Proposition 8], is $s$-weakly coupled for every $s>0$. The main goal of this section is to prove that system  satisfies the [Lie–Galerkin Stalking Condition]{}. As a consequence we obtain the following result, whose corollary is Theorem \[controllability\].

System is:

1.  [$L^{1}$-bounded]{}  approximately simultaneously controllable,

2.  approximately simultaneously controllable in $H^s$ for every $s> 0$,

3.  a stalker.

Using classical identities for Legendre polynomials and trigonometric relations one can prove that $$\langle Y^{m}_{{\ell}} , B_{j} Y^{m'}_{{\ell}} \rangle = 0$$ for every $j=1,2,3$, and $m,m' \in \{-{\ell}-1, \ldots, {\ell}+1\}$.

Moreover $$\langle Y^{m}_{{\ell}} , B_{j} Y^{m'}_{{\ell}'} \rangle = 0$$ with $|{\ell}-{\ell}'|\geq 2$ for every $m\in \{-{\ell}-1, \ldots, {\ell}+1\}$, $m'\in \{-{\ell}'-1, \ldots, {\ell}'+1\}$, $j=1,2,3$. In order to prove that the [Lie–Galerkin Stalking Condition]{} is satisfied, we choose a reordering $(\phi_k)_{k\in {\mathbb{N}}}$ of the spherical harmonics in such a way that $$\{\phi_k\mid k=1,\dots,4{\ell}+4\}=\{Y^{-{\ell}}_{\ell},\ldots,Y^{\ell}_{\ell},Y^{-{\ell}-1}_{{\ell}+1},\ldots,Y^{{\ell}+1}_{{\ell}+1}\},$$ and we are left to prove that $$\mathrm{Lie} \mathcal{V}_{4{\ell}+4} = \mathfrak{su}(4{\ell}+4).$$ The characteristics spectral gap of the space $\mathcal{H}_{\ell}$ is $({\ell}+1)({\ell}+2) -{\ell}({\ell}+1)=2 ({\ell}+1)$. In particular $
(2 ({\ell}+1),1) , (2 ({\ell}+1),2)$, and $(2 ({\ell}+1),3)$ are in $\Xi_{4{\ell}+4}$.

Matrix representations
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Denote by $\mathcal{J}_{\ell}$ 