subalgebra $\mathfrak{L}$ in $\mathfrak{A}$, generated by the operators $\left\{ P^{m,\,n}_{m,\,n}\right\}_{m,\,n\geq 0}$. We denote by $\mathfrak{M}$ Abelian subalgebra in $\mathfrak{A}$, obtained from $\mathfrak{L}$ by addition to generators $P^{m,\,n}_{m,\,n}$ of all formal series in these generators. Note that $$\label{13-7.12}
\mathbb{N}_1=\bigoplus_{m,\,n\geq0} m P^{m,\,n}_{m,\,n},\quad
\mathbb{N}_2=\bigoplus_{m,\,n\geq0} n P^{m,\,n}_{m,\,n},\quad
I=\bigoplus_{m,\,n\geq0}  P^{m,\,n}_{m,\,n},$$ which means that an arbitrary element $u\in\mathfrak{M}$ can be written as $$\label{13-7.13}
u=\bigoplus_{k,\,l\geq 0}c_{k,\,l}P^{k,\,l}_{k,\,l}.$$ We show that $\mathfrak{M}$ is a maximal Abelian subalgebra in Lie algebra $\mathfrak{A}$. Indeed, suppose that there is some element $z$ of $\mathfrak{A}$ such that $$[z,X]=0,$$ for all $X\in \mathfrak{M}$. Choosing $X=P^{s,\,t}_{s,\,t}$ (see reasons above), we obtain for $z$ the equality , i.e. $$\label{13-7.15}
z=d_0I+d_1\mathbb{N}_1+d_2\mathbb{N}_2
+\sum_{k\geq 0}d^{(L)}_{k,\,t;\,k,\,t}\widehat{P}^{[k],\,t}_{[k],\,t}
+\sum_{l\geq 0}d^{(R)}_{s,\,l;\,s,\,l}\widehat{P}^{s,\,[l]}_{s,\,[l]}
+\sum_{k,\,l\geq 0}d^{(D)}_{k,\,l;\,k,\,l}P^{k,\,l}_{k,\,l}.$$ Taking into account and , the relation can be rewritten as $$z=\bigoplus_{k,\,l\geq0}\left(k+l+d^{(D)}_{k,\,l;\,k,\,l}\right)P^{k,\,l}_{k,\,l}
\bigoplus_{k\geq0}\left(\sum_{s=0}^k d^{(L)}_{s,\,t;\,s,\,t}\right)P^{k,\,t}_{k,\,t}
\bigoplus_{l\geq0}\left(\sum_{t=0}^l d^{(R)}_{s,\,t;\,s,\,t}\right)P^{s,\,l}_{s,\,l}.$$ In view of , this means that $z\in\mathfrak{M}$. Hence it follows that subalgebra $\mathfrak{M}$ is the maximal Abelian subalgebra in the Lie algebra $\mathfrak{A}$.

Note that $\mathfrak{M}$ contains the commutative subalgebra constructed in work [@N47]. This subalgebra is an extension of Koornwinder algebra [@N28] consisting of all differential operators of the variables $z,\,\overline{z}$, for which Chebyshev-Koornwinder polynomials are eigenfunctions.

Subalgebras and ideals in $\AG$. Representation $\AG$ in the form of semidirect sum
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Let 