On the classification of Fernando-Kac subalgebras

Let $g$ be a finite-dimensional complex semisimple Lie algebra and $M$ be a $g$-module. The Fernando-Kac subalgebra of $g$ associated to $M$ is the subset $g[M] ⊂ g$ of all elements $g ∈ g$ which act locally finitely on $M$. A subalgebra $l ⊂ g$ for which there exists an irreducible module $M$ with $g[M] = l$ is called a Fernando-Kac subalgebra of $g$. A Fernando-Kac subalgebra of $g$ is of finite type if in addition $M$ can be chosen to have finite Jordan-Holder $l$-multiplicities.
In 2007 I. Penkov conjectured a classification of all root Fernando-Kac subalgebras of finite type (a root subalgebra is a subalgebra that contains a Cartan subalgebra). We will discuss the proof of this conjecture for simple $g≃/ E8$. Furthermore, we will discuss possible generalizations of the conjecture to arbitrary subalgebras of semisimple Lie algebras, with emphasis on subalgebras of rank $≤2$.