On the splitting case of a semi-biplane construction

We consider the case where a particular incidence structure splits into two substructures. The incidence structure in question was used previously by the authors to construct semi-biplanes $sbp(k^2,k)$ or $sbp(k^2/2,k)$. A complete description of the two substructures is obtained. We also show that none of the three semi-biplanes, $sbp(18,6)$, can be described using this construction.