In group theory Schreier's technique provides a basis for a
subgroup of a free group. In this paper an analogue is developed
for free Lie algebras. It hinges on the idea of cutting a Hall set
into two parts. Using it, we show that proper subalgebras of finite
codimension are not finitely generated and, following M.~Hall,
that a finitely generated subalgebra is a free factor of a
subalgebra of finite codimension.