Commutation Relations in Yangian-type Algebras

In paper \cite{olshanski2024centralizer}, G.I. Olshanski showed that there are four stable formulas for the commutator in the universal enveloping algebra $U(\mathfrak{gl}(N, \Omega))$ (where $\Omega= \mathbb{C}^L$). In a recent paper \cite{olshanski2023remarks} Nikita Safonkin and G.I. Olshanski generalized these results for an arbitrary algebra $\Omega$. \par\noindent This paper addresses one of the problems posed in section 8 of \cite{olshanski2024centralizer}, namely finding a more explicit presentation of the formula. The question is of interest, because the formulas are used in a construction which generalizes that of Yangians $Y_d = Y(\mathfrak{gl}(d, \mathbb{C}))$. \par\noindent In this work a recursive procedure to compute the formulas is derived. Several results regarding the structure of the formulas made through computer experiments are formulated and proved.