Characterizations of gaussian distributions on simply connected nilpotent Lie groups and symmetric spaces

For simply connected nilpotent Lie groups, we show that a probability measure is gaussian in the sense of Bernstein (for a definition thereof which in a natural way involves non-commutativity) iff it is a gaussian measure in the classical sense concentrated on an abelian subgroup. Furthermore we carry over the Skitovic-Darmois theorem to symmetric spaces of non-compact type.