  This work introduces a new concept, the so-called Darboux family, which is
employed to determine, to analyse geometrically, and to classify up to Lie
algebra automorphisms, in a relatively easy manner, coboundary Liebialgebras on
real four-dimensional indecomposable Lie algebras. The Darboux family notion
can be consideredas a generalisation of the Darboux polynomial for a vector
field. The classification of $r$-matrices and solutions to classical
Yang-Baxter equations for real four-dimensional indecomposable Lie algebras is
also given in detail. Our methods can further be applied to general, even
higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix
representations of certain Lie algebras with a non-trivial center is developed.
