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This work introduces a new concept, the so-called Darboux family, which isemployed to determine, to analyse geometrically, and to classify up to Liealgebra automorphisms, in a relatively easy manner, coboundary Liebialgebras onreal four-dimensional indecomposable Lie algebras. The Darboux family notioncan be consideredas a generalisation of the Darboux polynomial for a vectorfield. The classification of $r$-matrices and solutions to classicalYang-Baxter equations for real four-dimensional indecomposable Lie algebras isalso given in detail. Our methods can further be applied to general, evenhigher-dimensional, Lie algebras. As a byproduct, a method to obtain matrixrepresentations of certain Lie algebras with a non-trivial center is developed.

Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras