  A subalgebra of a semisimple Lie algebra is wide if every simple module of
the semisimple Lie algebra remains indecomposable when restricted to the
subalgebra. From a finer viewpoint, a subalgebra is $\lambda$-wide if the
simple module of a semisimple Lie algebra of highest weight $\lambda$ remains
indecomposable when restricted to the subalgebra. A subalgebra is narrow if the
restriction of all non-trivial simple modules to the subalgebra have proper
decompositions. We determine necessary and sufficient conditions for regular
subalgebras of semisimple Lie algebras to be $\lambda$-wide. As a natural
consequence, we establish necessary and sufficient conditions for regular
subalgebras to be wide, a result which has already been established by
Panyushev for essentially all regular solvable subalgebras. Next, we show that
establishing whether or not a regular subalgebra of a simple Lie algebra is
wide does not require consideration of all simple modules. It is necessary and
sufficient to only consider the adjoint representation. Finally, we show that a
regular subalgebra of the special linear algebra $\mathfrak{sl}_{n+1}$ is
either narrow or wide; this property does not hold for non-regular subalgebras
of $\mathfrak{sl}_{n+1}$.
