  The purpose of this paper is twofold. Firstly, to emphasise that the class of
Lie algebras with chain lattices of ideals are elementary blocks in the
embedding or decomposition of Lie algebras with finite lattice of ideals.
Secondly, to show that the number of Lie algebras of this class is large and
they support other types of Lie structures. Beginning with general examples and
algebraic decompositions, we focus on computational algorithms to build Lie
algebras in which the lattice of ideals is a chain. The chain condition forces
gradings on the nilradicals of this class of algebras. Our algorithms yield to
several positive naturally graded parametric families of Lie algebras. Further
generalizations and other kind of structures will also be discussed.
