Realization of Jordan-Kronecker invariants by Lie algebras

We study what Jordan-Kronecker invariants of Lie algebras, introduced by A. V. Bolsinov and P. Zhang, are possible. We completely solve this problem in the Jordan and the Kronecker cases. We prove that any JK invariants that contain the Kronecker $3 \times 3$ block or several Kronecker $1 \times 1$ blocks are possible. For other JK invariants, with Kronecker indices $k_1, \dots, k_q$, we give a partial answer: all Jordan--Kronecker invariants with no more than $\sum_i k_i$ Jordan tuples with multiple maxima are possible; the Jordan--Kronecker invariants with more than $\sum_i k_i$ unique Jordan tuples with multiple maxima are impossible. We also desribe all JK invariants that can be realized by compatible Poisson brackets with non-constant eigenvalues