Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on ${\mathfrak{g}}^{\ast }$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$onafinite-dimensionalLiealgebra$\frak{g}$. 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 ${\mathfrak{g}\mathfrak{l}}_n$.