We establish perturbative solutions for problems involving resonances of the unper-turbed situation, and therefore the necessity of a normal form, in the general framework of Banach scale Lie algebras defined below. This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses heavily mould calculus and in particular a universal mould equation studied in [PS16]