Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g∗ related to an algebraic Nijenhuis operator N:g→g on a finite-dimensional Lie algebra g. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related to Borel subalgebras and a new proof of the complete integrability of the free rigid body system on gln.