Complex contact Lie groups and generalized complex Heisenberg groups

In this paper, we use root decomposition techniques to classify the complex contact Lie groups such that the Reeb vector field action on the Lie algebra is diagonalizable. These groups turn out to be isomorphic on the Lie algebra level to a particular type of generalized Heisenberg groups, namely the semi-direct product C2n×ΩCC2n×ΩC, where Ω   is the standard symplectic 2-form on C2nC2n.