Wells-type exact sequence and crossed extensions of algebras with bracket

We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight-term exact sequence in the cohomology of algebras with bracket.