  In this paper, we give Maurer-Cartan characterizations as well as a
cohomology theory for compatible Lie algebras. Explicitly, we first introduce
the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan
characterizations of compatible Lie algebras. Then we introduce a cohomology
theory of compatible Lie algebras and use it to classify infinitesimal
deformations and abelian extensions of compatible Lie algebras. In particular,
we introduce the reduced cohomology of a compatible Lie algebra and establish
the relation between the reduced cohomology of a compatible Lie algebra and the
cohomology of the corresponding compatible linear Poisson structures introduced
by Dubrovin and Zhang in their study of bi-Hamiltonian structures. Finally, we
use the Maurer-Cartan approach to classify nonabelian extensions of compatible
Lie algebras.
