Root Fernando-Kac subalgebras of finite type

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$ which act locally finitely on $M$. A subalgebra $\mathfrak{l}\subset\mathfrak{g}$ for which there exists an irreducible module $M$ with $\mathfrak{g}[M]=\mathfrak{l}$ is called a Fernando-Kac subalgebra of $\mathfrak{g}$. A Fernando-Kac subalgebra of $\mathfrak{g}$ is of finite type if in addition $M$ can be chosen to have finite Jordan-H\"older $\mathfrak{l}$-multiplicities. Under the assumption that $\mathfrak{g}$ 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 $\mathfrak{g}\neq E_8$.