Summary: Generalized Schouten, Frölicher-Nijenhuis and Frölicher-Richardson brackets are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie algebroids are introduced and discussed and the behaviour of the related graded Lie brackets under these lifts is studied. In the case of the canonical Lie algebroid on the tangent bundle, a new common generalization of the Frölicher-Nijenhuis and the symmetric Schouten brackets, as well as embeddings of the Nijenhuis-Richardson and the Frölicher-Nijenhuis bracket into the Schouten bracket, are obtained.