The argument shift method in universal enveloping algebra $U{\mathfrak{g}\mathfrak{l}}_d$

We prove the conjecture that allows one extend the argument shifting procedure from symmetric algebra $S{\mathfrak{g}\mathfrak{l}}_d$ of the Lie algebra ${\mathfrak{g}\mathfrak{l}}_d$ to the universal enveloping algebra $U{\mathfrak{g}\mathfrak{l}}_d$. Namely, it turns out that the iterated quasi-derivations of the central elements in $U{\mathfrak{g}\mathfrak{l}}_d$ commute with each other. Here quasi-derivation is a linear operator on $U{\mathfrak{g}\mathfrak{l}}_d$, constructed by Gurevich, Pyatov and Saponov. This allows one better understand the structure of $\mathit{\text{argument}}\mathit{\text{shift}}\mathit{\text{algebras}}$ (or $\mathit{\text{Mishchenko-Fomenko}}\mathit{\text{algebras}}$) in the universal enveloping algebra of ${\mathfrak{g}\mathfrak{l}}_d$.