Verification of the generalized hypothesis of Mishchenko–Fomenko for Lie algebras of small dimension

In the case of Lie algebras $\mathfrak{g}$ of small dimension $\leq 7$, an enhanced version of the Generalised argument shift conjecture is proved, namely, it is shown that for any element $a\in\mathfrak{g}^*$ on the dual space $\mathfrak{g}^*$ there is a complete set of polynomials in the bi-involution with respect to the standard Poisson-Lie bracket and the frozen argument bracket associated with the covector $a$.