Equipping algebroid A with an additional Poisson structure allows to build a standard Lie algebroid structure on \(A^*\) . Under appropriate assumptions, this structure can be written as a linear combination of two other algebroid structures. For any sections X and Y of A satisfying a suitable commutation relation, we construct a Lie algebroid structure, or even a family of Lie algebroid structures, on the dual vector bundle \(A^*\) . In particular, taking \(A=TM\) we get the Lie algebroids on the cotangent bundle determined by vector fields X and Y.