I) The following conditions are equivalent.
For a proof, we refer the reader to the two references. Kunen in  shows that that any of the four conditions implies the existence of an identity element. And Bol and Bruck  show that the four conditions are equivalent for loops.
The 16-element set of unit octonions over is an example of a nonassociative Moufang loop. Other examples appear in projective geometry, coding theory, and elsewhere.
 Kenneth Kunen, Moufang Quasigroups, J. Algebra 83 (1996) 231–234. (A preprint in PostScript format is available from Kunen’s website: http://www.math.wisc.edu/ kunen/moufang.psMoufang Quasigroups.)
 R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1958.
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