Let $(V,+)$ be an abelian group, and let $(G,\cdot)$ be a commutative subgroup of the automorphism group, ${\rm Aut}\,V$, of $(V,+)$. Since $(V,+)$ is abelian, the authors have access to the endomorphism ring, ${\rm End}\,V$, of $(V,+)$. Two assumptions are made: (1) Unique ``square roots'' are available for $(G,\cdot)$, that is, $g\mapsto g\cdot g$ is a permutation of $G$, which guarantees that to each $g$ in $G$ there corresponds a unique $\sqrt{g}$ in $G$ such that $\sqrt{g}\cdot\sqrt{g}=g$. (2) Each $1+g$ in ${\rm End}\,V\ (g$ in $G$ and 1 the identity automorphism of $(V,+))$ is actually in ${\rm Aut}\,V$. With assumptions (1) and (2) (and frequently with the tacit assumption that $|V|>1)$, the authors succeed in imposing an invariant reflection structure on the Cartesian product $P=G\times V$ in the sense of H. J. Karzel\ [Discrete Math. {\bf 208/209} (1999), 387--409; MR1725545 (2000i:51004)], and also introduce an operation ``+'' on $P=G\times V$ so that $(P,+)$ is a $K$-loop (see [A. Kreuzer\ and H. Wefelscheid, Results Math. {\bf 25} (1994), no.~1-2, 79--102; MR1262088 (95a:20072)] for information on $K$-loops). \par The authors introduce conditions on $(G,\cdot)$ which enable them to establish an incidence bundle on $(P,+)$. They do so in such a way that the $K$-loop $(P,+)$ is an incidence fibered loop as described by E. Zizioli\ [J. Geom. {\bf 30} (1987), no.~2, 144--156; MR0918823 (89a:51038)].