  We establish an improved upper estimate on dimension of any solvable algebra
s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we
consider Levi decomposable algebras with a given nilradical n and investigate
restrictions on possible Levi factors originating from the structure of
characteristic ideals of n. We present a new perspective on Turkowski's
classification of Levi decomposable algebras up to dimension 9.
