Let K be a field of characteristic zero. Let B be a semisimple Lie algebra over K, V a finite dimensional B-module over K. Let \({\mathcal I}\) denote the free Lie algebra generated by the vector space V and \(B\oplus {\mathcal I}\) be the split extension of B in the sense of N. Jacobson [Lie algebras (1962; Zbl 0121.275)]. An ideal \(I\subseteq B\oplus {\mathcal I}\) is said to be regular if \({\mathcal I}^ 2\supseteq I\supseteq {\mathcal I}^ m\) for some m.
The following results have been given in this paper: For each regular ideal \(I\subseteq B\oplus {\mathcal I}\), the quotient algebra \(B\oplus {\mathcal I}/I={\mathcal L}\) which is a finite dimensional Lie algebra has a nilpotent radical R such that \(V=R/R^ 2\) (can be viewed as a B-module) and \(B={\mathcal L}/R.\)
Conversely, let \({\mathcal L}\) be a finite dimensional Lie algebra with nilpotent radical R such that \(V=R/R^ 2\) can be viewed as a B-module where \(B={\mathcal L}/R\). Then there is a regular ideal \(I\subseteq B\oplus {\mathcal I}\) such that \({\mathcal L}\cong B\oplus {\mathcal I}/I\).