New techniques for calculation of Jordan-Kronecker invariants for Lie algebras and Lie algebra representations

We introduce two novel techniques that simplify calculation of Jordan-Kronecker invariants for a Lie algebra $\mathfrak{g}$ and for a Lie algebra representation $\rho$. First, the stratification of matrix pencils under strict equivalence puts restrictions on the Jordan-Kronecker invariants. Second, we show that the Jordan-Kronecker invariants of a semi-direct sum $\mathfrak{g} \ltimes_{\rho} V$ are sometimes determined by the Jordan-Kronecker invariants of the dual Lie algebra representation $\rho^*$.