  In a recent work by two of us the argument shift method was extended from the
symmetric algebra ${\rm S}({\mathfrak g})$ of the general linear Lie algebra
${\mathfrak g}$ to the universal enveloping algebra ${\rm U}({\mathfrak g})$.
We show in this paper that some features of this 'quantum argument shift
method' can be applied to the remaining classical matrix Lie algebras
${\mathfrak g}$. We prove that a single application of the quasi-derivation to
central elements of ${\rm U}({\mathfrak g})$ yields elements of the
corresponding quantum Mishchenko-Fomenko subalgebra. We show that generators of
this subalgebra can be obtained by iterated application of the quasi-derivation
to generators of the center of ${\rm U}({\mathfrak g})$.
