Lie algebroids generated by cohomology operators

By studying the Frölicher-Nijenhuis decomposition of cohomology operators (that is,derivations $D$ of the exterior algebra $\Omega (M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$),we describe new examples of Lie algebroid structures on the tangent bundle $TM$(and its complexification $T^{\mathbb{C}}M$)constructed from pre-existing geometric ones such as foliations,complex, product or tangent structures.We also describe a class of Lie algebroids on tangent bundles associated to idempotent endomorphisms with nontrivial Nijenhuis torsion.