we have that also $E_{({\ell},m),({\ell}+1,m+1)}$ is in $L$ for every $m= -{\ell},\ldots,{\ell}$, proving . It then follows from that each $E_{j,k}$ is in $L$.

If now we replace by $$\frac{B_{1}+J_{i}(B_{2})}{2} = -\sum_{m=-{\ell}}^{{\ell}} q_{{\ell},m} F_{({\ell},m),({\ell}+1,m-1)}\in L,$$ or $$\frac{B_{1}-J_{i}(B_{2})}{2} = \sum_{m=-{\ell}}^{{\ell}} q_{{\ell},-m} F_{({\ell},m),({\ell}+1,m+1)}\in L,$$ we obtain from the arguments above that $F_{({\ell},m),({\ell}+1,m-1)}$ and $F_{({\ell},m),({\ell}+1,m+1)}$ are in $L$ for every $m= -{\ell},\ldots,{\ell}$. The relations and allow then to conclude that $L=\mathfrak{su}(4{\ell}+4)$.

Proof of Theorem \[metalemma\] {#sec:prova1}
==============================

Time-reparametrization {#s-time}
----------------------

Up to replacing each $B_j$ by $\delta B_j$, we can assume that $\delta=1$.

For every piecewise constant function $z$ such that $z(t)\geq 1$ for every $t$, we consider the time-reparametrization $$\label{eq:main-repar0}
\frac{d\psi}{dt}(t)=(z(t)A +  u_1(t)z(t)B_1+ \cdots + u_p(t)z(t) B_p) \psi(t)$$ of system . Each $u_j(t)z(t)$ belongs to the time-varying set $z(t)U_j$.

If $u_{1}, \ldots, u_{p}$ are control laws in  then the corresponding controls in  are their time-reparametrizations $\tilde u_{j}(s) = u_{j}(t(s))$ with $t(s)=\int_{0}^{s}z(\tau)d\tau$, $j=1,\ldots,p$. By restricting the range of available controls and setting $v_{j}(t)=u_{j}(t)z(t)$, we can focus our attention to trajectories of $$\label{eq:main-repar}
\frac{d\psi}{dt}(t)=(z(t)A+v_1(t)B_1+ \cdots + v_p(t) B_p) \psi(t),\quad z(t)\geq 1,\quad v(t)=(v_1(t),\dots,v_p(t))\in U.$$ Each solution of with $z$ and $v$ piecewise constant is the time-reparametrization of a solution of  with piecewise constant controls (but 