A root space decomposition for finite vertex algebras

Let L be a Lie pseudoalgebra, a is an element of L. We show that, if a generates a (finite) solvable subalgebra S = < a > subset of L, then one may find a lifting (a) over bar is an element of S of [a] is an element of S/S' such that <(a) over bar > is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U x N, where U is a subalgebra of V whose underlying Lie conformal algebra U-Lie is a nilpotent self-normalizing subalgebra of V-Lie, and N = V-[infinity] is a canonically determined ideal contained in the nilradical Nil V.