  Levi's theorem decomposes any arbitrary Lie algebra over a field of
characteristic zero, as a direct sum of a semisimple Lie algebra (named Levi
factor) and its solvable radical. Given a solvable Lie algebra $R$, a
semisimple Lie algebra $S$ is said to be a Levi extension of $R$ in case a Lie
structure can be defined on the vector space $S\oplus R$. The assertion is
equivalent to $\rho(S)\subseteq \mathrm{Der}(R)$, where $\mathrm{Der}(R)$ is
the derivation algebra of $R$, for some representation $\rho$ of $S$ onto $R$.
Our goal in this paper, is to present some general structure results on
nilpotent Lie algebras admitting Levi extensions based on free nilpotent Lie
algebras and modules of semisimple Lie algebras. In low nilpotent index a
complete classification will be given. The results are based on linear algebra
methods and leads to computational algorithms.
