2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74922We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on $\frak{g}^*$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$ on a finite-dimensional Lie algebra $\frak{g}$. 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 $\frak{gl}_n$.10 pages, references addedDifferential Geometry53D17,53D20, 37J35text