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

Representation Theory;Quantum Algebra