Connectivity of abelian Cayley graphs containing no \(K_4\) (Q1813994)

It was proved in [\textit{W. Mader}, Arch. Math. 21, 331--336 (1970; Zbl 0201.56804)] that every connected, vertex-transitive, finite graph without a subgraph \(K_4\) has vertex-connectivity equal to the degree of the vertices. It is proved now that this remains true for abelian Cayley digraphs without a subdigraph \(\vec K_4\), if they have no symmetric edge, where \(\vec K_4\) means any orientation of \(K_4\). It is not true for all connected, finite, abelian Cayley digraphs without \(\vec K_4\), but the exceptions are characterized.