Geodesic
The upper triangular algebra loop of degree $4$
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 4, pp. 457-470.

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A natural loop structure is defined on the set $U_4$ of unimodular upper-triangular matrices over a given field. Inner mappings of the loop are computed. It is shown that the loop is non-associative and nilpotent, of class 3. A detailed listing of the loop conjugacy classes is presented. In particular, one of the loop conjugacy classes is shown to be properly contained in a superclass of the corresponding algebra group.
Classification : 20N05
Keywords: algebra group; quasigroup; loop; supercharacter; fusion scheme 
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     title = {The upper triangular algebra loop of degree $4$},
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Johnson, K. W.; Munywoki, M.; Smith, Jonathan D. H. The upper triangular algebra loop of degree $4$. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 4, pp. 457-470. http://geodesic.mathdoc.fr/item/CMUC_2014__55_4_a2/