On the maximal finite-dimensional Lie algebras with given nilradical

We study the set of finite-dimensional Lie algebras with fixed nilradical (in the capacity of which any nilpotent Lie algebra may serve). We prove an exact estimate for dimensions of Lie algebras from this set. We also show that there may exist several Lie algebras in this set, possessing the maximal dimension. Proofs are based on a concept of algebraic splitting for finite-dimensional Lie algebras.