  By studying the Fr\"olicher-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.
