Root Fernando–Kac subalgebras of finite type

Let g be a finite-dimensional 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–Hölder l-multiplicities. Under the assumption that g is simple, I. Penkov has conjectured an explicit combinatorial criterion describing all Fernando–Kac subalgebras of finite type which contain a Cartan subalgebra. In the present paper we prove this conjecture for g≄E8.