Investigation of algebra $\AG$
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Commutation relations between generators of the algebra $\AG$
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For the convenience of further considerations we shall divide the set of generators of the algebra $\AG$ on 4 subset.

$$\text{
\begin{tabular}{|c|c|c|}
\hline
\text{type}&\text{generators}&\text{symbolic notation}\\
\hline
$I$& \rule[-7pt]{0pt}{25pt}$\mathbb{I},\, \mathbb{N}_1,\, \mathbb{N}_2$& $A_I=A$ \\
\hline
$II$& \rule[-7pt]{0pt}{25pt}$\aspm,\,\arpm$ & $A_{II}=B$\\
\hline
$III_{left}$& \rule[-7pt]{0pt}{25pt}$\hPnkm{[k],\,l}{[m]}{n}$ & $A_{III}^{left}=C^{l}$\\
\hline
$III_{right}$& \rule[-7pt]{0pt}{25pt}$\hPnkm{k,\,[l]}{m}{[n]}$  &$A_{III}^{right}=C^{r}$ \\
\hline
$IV$& \rule[-7pt]{0pt}{25pt}$\Pnkm{k,\,l}{m}{n}$&$A_{IV}=D$ \\
\hline
\end{tabular}
}$$

From relations - it follows that $$\label{13-7.2}
[A_1,A_2]=0,\quad [A,B_1]=B_2,\quad [A,C^l_1]=C^l_2,\quad [A,C^r_1]=C^r_2,\quad [A,D_1]=D_2;$$ $$\label{13-7.3}
[B_1,B_2]=C^l+C^r+D,\quad [B,C^l_1]=C^l_2+C^l_3,\quad [B,C^r_1]=C^r_2+C^r_3,
\quad [B,D_1]=D_2+D_3;$$ $$\label{13-7.4}
\left\{
\begin{aligned}
&[C^l_1,C^l_2]=C^l_3+C^l_4,\quad [C^r_1,C^r_2]=C^r_3+C^r_4,\quad [C^l,C^r]=D_1+D_2,\\
&\qquad\qquad\qquad\qquad[C^l,D_1]=D_2+D_3,\quad [C^r,D_1]=D_2+D_3;
\end{aligned}
\right.$$ $$\label{13-7.5}
[D_1,D_2]=D_3+D_4.$$

Center $\mathfrak{Z}(\mathfrak{A})$ of the algebra $\AG$
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An element $z$ belongs to the center $\mathfrak{Z}(\mathfrak{A})$ of the algebra $\AG$ if and only if this element commutes with all generators $X$ of the algebra $\AG$: $$\label{13-7.6}
    [z,X]=0.$$ The element X can be represented in the following form $$\begin{gathered}
\label{13-7.7}
    z=d_0I+d_1\mathbb{N}_1+d_2\mathbb{N}_2+d^+_s\asp+d^-_s\asm+d^+_r\arp+d^-_r\arm+\\[3pt]
    \sum_{k,\,l,\,m,\,n\geq 0}\left(d^{(L)}_{k,\,l;\,m,\,n}\widehat{P}^{[k],\,l}_{[m],\,n}+
    d^{(R)}_{k,\,l;\,m,\,n}\widehat{P}^{k,\,[l]}_{m,\,[n]}\right)+
    \sum_{k,\,l,\,m,\,n\geq 0}d^{(D)}_{k,\,l;\,m,\,n}P^{k,\,l}_{m,\,n}.\qquad\end{gathered}$$

Choose $X=P^{s,\,t}_{s,\,t}$ in . From the commutation relations - we get

$$\begin{aligned}
[I, P^{s,\,t}_{s,\,t}]=0,\quad [\mathbb{N}_1, P^{s,\,t}_{s,\,t}]=0,\quad
[\mathbb{N}_2, P^{s,\,t}_{s,\,t}]=0;\qquad\qquad\qquad\qquad\qquad \label{13-7.8a} \\
\left[ \aspm , P^{s,\,t}_{s,\,t} \right]=
\frac{1}{\sqrt{10}}\left(P^{s,\,t}_{s\mp 1,\,t\pm 1}-P^{s\pm 1,\,t\mp 1}_{s,\,t}\right)\neq 0;
\qquad\qquad\qquad\qquad\qquad \label{13-7.8b}\\
\left[ \arpm , P^{s,\,t}_{s,\,t} \right]=
\sqrt{\frac{2}{5}}\left(P^{s,\,t}_{s\pm 1,\,t\pm 1}-P^{s\mp 1,\,t\mp 1}_{s,\,t}\right)\neq 0;
\qquad\qquad\qquad\qquad\qquad \label{13-7.8c}\\
\left[\widehat{P}^{[k],\,l}_{[m],\,n},\,P^{s,\,t}_{s,\,t} \right]=
