Supplements on the theory of exponential Lie groups

A Lie group with surjective exponential function is called exponential. There are presented supplying results on the theory of exponential Lie groups. An additional criterion for (Malʹcev) splittable exponential Lie groups is presented as well as an additional necessary condition for solvable exponential Lie groups. Both the conditions have the advantage that they are well practicable. Moreover, it is shown that there exist simple non-linear exponential Lie groups. The general case is also considered: There are given some conditions concerning the Malʹcev splittable radical. At the end of the paper, one can find some counterexamples to some conjectures concerning exponential Lie groups.