In his paper [4] G. A. Hunt described the generating functional of a convolution semigroup on a Lie group (representation theorem). Furthermore he has shown that conversely certain functionals determine convolution semigroups (generation theorem). V~. Hazod [2] extended these results to arbitrary locally compact groups by Lie gToup approximation. In [5] the author gave an algebraic characterization of the generating functionals with the aim to present the results of Hazod in a closed form. But the full power of this method has yet not been exhausted in the paper [5]. In the following discussion we present a new proof of the generation theorem for convolution semigroups which is adapted to the framework given in [5]. In addition the demonstration uses only few results of the Hille-u theory. The essential idea is that we first prove the theorem for Lie projective groups (the Lie group case is taken for granted), and then extend it to arbitrary groups.