Normal forms in a cyclically graded Lie algebra

In a 1999 paper published in JSC the second author and Martin Kummer proved “that in all but one case the normal form of a real or complex Hamiltonian matrix which is irreducible and appropriately normalized can be computed by Lie series methods in formally the same manner as one computes the normal form of a nonlinear Hamiltonian function”. In this paper we handle the final case using an extension of those Lie series methods developed by the present authors. Those extended methods make use of spectral sequences and are developed in considerable generality, i.e., their applicability is by no means restricted to the Hamiltonian context of this paper. A specific example is worked out in detail.