A root space decomposition for finite vertex algebras

Let L be a Lie pseudoalgebra, a in L. We show that, if a generates a (finite) solvable subalgebra S=<a>, then one may find a lifting a in S of [a] in S/S such that <a> 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 + 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 is a canonically determined ideal contained in the nilradical Nil V.