AbstractFor a solvable Lie group G the surjectivity of the exponential function expG is equivalent to the connectedness of the near‐Cartan subgroups and to the connectedness of the centralizers in a Cartan subgroup of all nilpotent elements in its Lie algebra g. Furthermore, these conditions are satisfied if and only if for all elements g ϵ G there is an x ϵ g with g = expG x in which expG is regular.