  In this paper we introduce the notion of pure non-characteristically
nilpotent Lie algebra and under a condition we prove that a complex maximal
extension of a finite-dimensional pure non-characteristically nilpotent Lie
algebra is isomorphic to a semidirect sum of the nilradical and its maximal
torus. We also prove that such solvable Lie algebras are complete and we
specify a subclass of the maximal solvable extensions of pure
non-characteristically nilpotent Lie algebras that have trivial cohomology
group. Some comparisons with the results obtained earlier are given.
