and using the bracket relations , we have $$\begin{aligned}
{\operatorname{ad}}^{2j}_{B_{3}} J_{i}(B_{3}) &= [B_{3},[B_{3},{\operatorname{ad}}^{2j-2}_{B_{3}} J_{i}(B_{3}) ] ] \\
 &= (-1)^{j}  2^{2j} \sum_{m=-{\ell}}^{{\ell}} p_{{\ell},m}^{2j+1} E_{({\ell},m),({\ell}+1,m)}.\end{aligned}$$

By invertibility of the Vandermonde matrix and since $p_{{\ell},m} \neq p_{{\ell},n}$ for every $n \neq m,-m$, it follows $$\label{eq:verticali}
E_{({\ell},-m),({\ell}+1,-m)}+E_{({\ell},m),({\ell}+1,m)}\in L\qquad\mbox{ for } m=0,\dots,{\ell}.$$

In particular $E_{({\ell},0),({\ell}+1,0)}\in L$. The double bracket of $$\label{unaaa}
 \frac{B_{2} - J_{i}(B_{1})}{2} = \sum_{m=-{\ell}}^{{\ell}} q_{{\ell}, m} E_{({\ell},m),({\ell}+1,m-1)}\in L,$$ with $E_{({\ell},0),({\ell}+1,0)}$ is easily computed using and and gives $$\begin{aligned}
\Big[\Big[ \sum_{m=-{\ell}}^{{\ell}} q_{{\ell}, m} E_{({\ell},m),({\ell}+1,m-1)}, E_{({\ell},0),({\ell}+1,0)}\Big],E_{({\ell},0),({\ell}+1,0)}\Big]&= 
 -q_{{\ell},1}[E_{({\ell},0),({\ell},1)},E_{({\ell},0),({\ell}+1,0)}] \\
 &\quad  -q_{{\ell},0} [E_{({\ell}+1,-1),({\ell}+1,0)},E_{({\ell},0),({\ell}+1,0)}]\\
&=  q_{{\ell},0}E_{({\ell},0),({\ell}+1,-1)}+q_{{\ell},1} E_{({\ell},1),({\ell}+1,0)} \in L.\end{aligned}$$

Define $Q_0= q_{{\ell},0}E_{({\ell},0),({\ell}+1,-1)}+q_{{\ell},1} E_{({\ell},1),({\ell}+1,0)}$ and, similarly, $Q_m= q_{{\ell},-m}E_{({\ell},-m),({\ell}+1,-m-1)} + q_{{\ell},m+1} E_{({\ell},m+1),({\ell}+1,m)}$ for $0<m<{\ell}$, $Q_{\ell}= q_{{\ell},-{\ell}}E_{({\ell},-{\ell}),({\ell}+1,-{\ell}-1)}$. In particular $ B_{2} - J_{i}(B_{1}) = 2\sum_{m=0}^{{\ell}} Q_m$.

Using again and , we have $$\begin{aligned}
\Big[\Big[ \sum_{m=k}^{\ell}Q_m, E_{({\ell},-k),({\ell}+1,-k)}+ E_{({\ell},k),({\ell}+1,k)} \Big], E_{({\ell},-k),({\ell}+1,-k)}+E_{({\ell},k),({\ell}+1,k)} \Big] 
= Q_{k},$$ for $k=1,\dots,{\ell}$. By recurrence on $k$ and because of , it follows that $Q_k\in L$ for $k=0,\dots,{\ell}$.

Now, since $Q_{\ell}/q_{{\ell},-{\ell}}=E_{({\ell},-{\ell}),({\ell}+1,-{\ell}-1)}$ is in $L$, then $$\begin{aligned}
\mathrm{ad}_{E_{({\ell},-{\ell}),({\ell}+1,-{\ell}-1)}}^2 (E_{({\ell},-{\ell}),({\ell}+1,-{\ell})}+E_{({\ell},{\ell}),({\ell}+1,{\ell})})&=
  -E_{({\ell},-{\ell}),({\ell}+1,-{\ell})}\in L,$$ which, in turns, implies that $$\begin{aligned}
\mathrm{ad}_{E_{({\ell},-{\ell}),({\ell}+1,-{\ell})}}^2 (Q_{{\ell}-1})&=
 -q_{{\ell},-{\ell}}E_{({\ell},-{\ell}+1),({\ell}+1,-{\ell})} \in L.$$ Iterating the argument, $E_{({\ell},m),({\ell}+1,m)}$ and $E_{({\ell},m),({\ell}+1,m-1)}$ are in $L$ for every $m= -{\ell},\ldots,{\ell}$.

Developing the same argument as above replacing by $$\frac{B_{2} + J_{i}(B_{1})}{2} = \sum_{m=-{\ell}}^{{\ell}} q_{{\ell},-m} E_{({\ell},m),({\ell}+1,m+1)}\in L,$$ 