Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

If G is a finite group whose integral group ring Z[G] has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring Q[G] have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central units in Z[G] cannot exist. With this, we are able to greatly simplify the argument that characterizes those 3-groups with integral group ring having MJD. Furthermore, we show that if G is a nonabelian semidirect product of the form Cp⋊C3k, with prime p > 7 and with the cyclic 3-group acting like a group of order 3, then Z[G] does not have MJD.