Cyclic wide subalgebras of semisimple Lie algebras

Let s r be a Levi decomposable Lie algebra, with Levi factor s, and radical r. A module V of s r is cyclic indecomposable if it is indecomposable and the quotient module V /r V is a simple s-module. A Levi decomposable subalgebra of a semisimple Lie algebra is cyclic wide if the restriction of every simple module of the semisimple Lie algebra to the subalgebra is cyclic indecomposable. We establish a condition for a regular Levi decomposable subalgebra of a semisimple Lie algebra to be cyclic wide. Then, in the case of a regular Levi decomposable subalgebra whose radical is an ad-nilpotent subalgebra, we show that the condition is necessary and sufficient for the subalgebra to be cyclic wide. All Lie algebras, and modules in this article are finite-dimensional, and over the complex numbers.