Jordan decompositions in alternative loop rings (Q1851416)

It is well known that, if \(G\) is a finite group and \(K\) a field of characteristic zero, then every element \(\alpha\) of the group ring \(KG\) possesses a decomposition as a sum of commuting semisimple and nilpotent elements, and, if \(\alpha\) is a unit, then this yields an expression for \(\alpha\) as a product of a semisimple and unipotent element. In this paper the question of the existence of such decompositions when the group is replaced by a finite \(RA\) loop \(L\) is considered. One result that can be stated simply is that \(ZL\) has additive Jordan decomposition if and only if \(L\) is a Hamiltonian Moufang RA 2-loop.