A root space decomposition for finite vertex algebras

Let L be a Lie pseudoalgebra, a ∈ L. We show that, if a generates a (finite) solvable subalgebra S = ⟨a⟩ ⊂ L, then one may find a lifting a ̄ ∈ S of [a] ∈ 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 ULie is a nilpotent self-normalizing subalgebra of V Lie, and N = V [∞] is a canonically determined ideal contained in the nilradical Nil V .