  Let $\mathfrak{s}$ $\ltimes$ $\mathfrak{r}$ be a Levi decomposable Lie
algebra, with Levi factor $\mathfrak{s}$, and radical $\mathfrak{r}$. A module
$V$ of $\mathfrak{s}$ $\ltimes$ $\mathfrak{r}$ is cyclic indecomposable if it
is indecomposable and the quotient module $V /\mathfrak{r}\cdot V$ is a simple
$\mathfrak{s}$-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.
