Journal of Lie Theory

We use Kronecker's method to construct systems of functions in bi-involution with respect to two Poisson brackets: the canonical one and the bracket with frozen argument $A\in \mathfrak{g}$. For the Lie algebras $\mathfrak {sl}_n$ and $\mathfrak {sp}_{2n}$, we construct complete systems of functions in bi-involution for any $A \in \mathfrak{g}$. For the Lie algebras $\mathfrak {so}_{2n+1}$ and $\mathfrak {so}_{2n}$, we describe elements $A$ such that we can construct a complete system of functions in bi-involution and the elements $A$ such that we can construct the Kronecker part of a complete system of functions in bi-involution. Also, we prove that the constructed functions freely generate some limits of Mishchenko-Fomenko subalgebras. Finally, for the Lie algebras $\mathfrak {sl}_n$ and $\mathfrak {sp}_{2n}$, we show that the Kronecker indices are the same for all elements $A$ in any given sheet, while for the Lie algebras $\mathfrak {so}_{2n}$ and $\mathfrak {so}_{2n+1}$, we give examples of sheets such that this is not true.