The main aim of the paper is to prove still another version of the Lvy--Khintchine decomposition of conditionally positive
definite functions on a general locally compact Abelian group. The exposition is based on the two-cones theorem proved by
N. Drumm in 1976. Application of the main result to the Euclidean group shows the novelty of the approach.
A notion of Gaussian hemigroup is introduced and its relationship with the Gauss condition is studied. Moreover, a Lvy-type
martingale characterization is proved for processes with independent (not necessarily stationary) increments satisfying the
Gauss condition in a compact Lie group. The characterization is given in terms of a faithful finite dimensional representation
of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups
associated with the increment processes.