Economizing Brackets to Define Filiform Lie Algebras

It is usual to define the law of a Lie algebra by giving explicitly the nonzero brackets between the elements of one of its bases.However, this paper shows that it is possible to reduce significatively the number of the brackets which are normally indicated when defining a filiform Lie algebra.Indeed, two particular families of brackets are considered and it is proved that the algebra can be defined by using only the elements of anyone of them.