& \left[\mathbb{N}_2,\widehat{P}^{[k],\,l}_{[m],\,n}\right]=(n-l)
\widehat{P}^{[k],\,l}_{[m],\,n};\\[5pt]
& \left[\mathbb{N}_2,\widehat{P}^{k,\,[l]}_{m,\,[n]}\right]=(n-l)
\widehat{P}^{k,\,[l]}_{m,\,[n]};
\end{aligned}
\right.$$

$$\label{13-6.14}
\left\{
\begin{aligned}
&\left[\asm,\asp\right]=
\frac{1}{10}\left(\hPnkm{[0],\,0}{[0]}{0}-\hPnkm{0,\,[0]}{0}{[0]}\right),\\[5pt]
&\left[\aspm,\arpm\right]=\pm\frac{1}{5}\hPnkm{0,\,[\mp 1]}{0}{[\pm 1]},\\[5pt]
&\left[\asmp,\arpm\right]=\pm\frac{1}{5}\hPnkm{[\mp 1],\,0}{[\pm 1]}{0},\\[5pt]
&\left[\aspm,\widehat{P}^{[k],\,l}_{[m],\,n}\right]=\frac{1}{\sqrt{10}}
\left(\widehat{P}^{[k],\,l}_{[m\mp 1],\,n\pm 1} -
\widehat{P}^{[k\pm 1],\,l\mp 1}_{[m],\,n}\right),\\[5pt]
& \left[\aspm,\widehat{P}^{k,\,[l]}_{m,\,[n]}\right]=\frac{1}{\sqrt{10}}
\left(\widehat{P}^{k,\,[l]}_{m\mp 1,\,[n\pm 1]}-
\widehat{P}^{k\pm 1,\,[l\mp 1]}_{m,\,[n]}\right),\\[5pt]
& \left[\asmp,\Pnkm{k,\,l}{m}{n}\right]=\frac{1}{\sqrt{10}}
\left(\Pnkm{k,\,l}{m\pm 1}{n\mp 1}- \Pnkm{k\mp 1,\,l\pm 1}{m}{n}\right).
\end{aligned}
\right.$$

$$\label{13-6.15}
\left\{
\begin{aligned}
& [\arm,\arp]=\frac{2}{5}\left(\hPnkm{[0],\,0}{[0]}{0}-
\hPnkm{0,\,[0]}{0}{[0]}-P^{0,\,0}_{0,\,0}\right);\\[5pt]
& \left[\arpm,\widehat{P}^{[k],\,l}_{[m],\,n}\right]=\sqrt{\frac{2}{5}}
\left(\widehat{P}^{[k],\,l}_{[m\pm 1],\,n\pm 1}-
\widehat{P}^{[k\mp 1],\,l\mp 1}_{[m],\,n}\right)\\[5pt]
& \left[\arpm,\widehat{P}^{k,\,[l]}_{m,\,[n]}\right]=\sqrt{\frac{2}{5}}
\left(\widehat{P}^{k,\,[l]}_{m\pm 1,\,[n\pm 1]}-
\widehat{P}^{k\mp 1,\,[l\mp 1]}_{m,\,[n]}\right)\\[5pt]
& \left[\armp,\Pnkm{k,\,l}{m}{n}\right]=\sqrt{\frac{2}{5}}\left(
\Pnkm{k,\,l}{m\mp 1}{n\mp 1}- \Pnkm{k\pm 1,\,l\pm 1}{m}{n}\right).
\end{aligned}
\right.$$

$$\label{13-6.16}
\left\{
\begin{aligned}
& \left[\widehat{P}^{[k],\,l}_{[m],\,n},\widehat{P}^{[q],\,s}_{[u],\,v}\right]=
\left(\widehat{P}^{[u+v-k-l-q],\,s}_{[m],\,n}
-\widehat{P}^{[m+n-q-s-k],\,l}_{[u],\,v}\right)\\[5pt]
& \left[\widehat{P}^{[k],\,l}_{[m],\,n},\widehat{P}^{q,\,[s]}_{u,\,[v]}\right]=
\left(P^{q,\,l-v+s}_{u-k+m,\,n}-P^{q-m+k,\,l}_{u,\,n-s+v}\right)\\[5pt]
& \left[\widehat{P}^{k,\,[l]}_{m,\,[n]},\widehat{P}^{q,\,[s]}_{u,\,[v]}\right]=
\left(\widehat{P}^{q,\,[-s-k-l+u+v]}_{m,\,[n]}-
\widehat{P}^{k,\,[-l+m+n-q-s]}_{u,\,[v]}\right)
\end{aligned}
\right.$$

$$\label{13-6.17}
\left\{
\begin{aligned}
& \left[\widehat{P}^{[k],\,l}_{[m],\,n},P^{q,\,s}_{u,\,v}\right]=
\left(P^{q,\,s}_{u+v+m-k-l,\,n}-P^{q+s+k-m-n,\,l}_{u,\,v}\right);\\[5pt]
& \left[\widehat{P}^{k,\,[l]}_{m,\,[n]},P^{q,\,s}_{u,\,v}\right]=
\left(P^{q,\,s}_{m,\,u+v+n-k-l}-P^{k,\,q+s+l-m-n}_{u,\,v}\right).
\end{aligned}
\right.$$

$$\label{13-6.18}
\left[\Pnkm{k,\,l}{m}{n},\Pnkm{s,\,t}{u}{v}\right]=\delta_{k,\,u}\delta_{l,\,v}\Pnkm{s,\,t}{m}{n}-
\delta_{m,\,s}\delta_{n,\,t}\Pnkm{k,\,l}{u}{v}.$$

generate the ideal $I$ of the algebra $\widehat{\AG}$.

We define algebra $\AG$ as the factor algebra $$\AG=\widehat{\AG}/I.$$ The algebra $\AG$ is the infinite-dimensional associative algebra generated by the operators $$\label{13-6.20}
\mathbb{I},\, \mathbb{N}_1,\, \mathbb{N}_2,\, \aspm,\, \arpm,\,
\widehat{P}^{k,\,[l]}_{m,\,[n]},\, \widehat{P}^{[k],\,l}_{[m],\,n},\, \Pnkm{k,\,l}{m}{n},$$ satisfying commutation relations -. (Recall that it is assumed that the indexes $k,\,l,\,m,\,n\geq0$, and if at least one of the indices of operators $P$ and $\widehat{P}$ is negative, then the corresponding operator is equal to zero).

The algebras $\widehat{\AG}$ and $\AG$ are Lie algebras because one can check that for generators are fulfilled Jacobi identities.

