We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

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In the case when M is equipped with a bi-Hamiltonian structure (M, π 1 , π 2) we show how to construct family of Poisson structures on the tangent bundle T M to a Poisson manifold. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.

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