Bruck decomposition for endomorphisms of quasigroups

In the year 1944 R. H. Bruck has described a very general construction method which he called the extension of a set by a quasigroup. We use it to construct a class of examples for LF-quasigroups in which the image of the map \(e(x)=x\backslash x\) is a group. More generally, we consider the variety of quasigroups which is defined by the property that the map \(e\) is an endomorphism and its subvariety where the image of the map \(e\) is a group. We characterize quasigroups belonging to these varieties using their Bruck decomposition with respect to the map \(e\).