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We start by describing all the varieties of loops Q that can be defined by autotopisms α x , x Q , where α x is a composition of two triples, each of which becomes an autotopism when the element x belongs to one of the nuclei. In this way we obtain a unifying approach to Bol, Moufang, extra, Buchsteiner and conjugacy closed loops. We re-prove some classical facts in a new way and show how Buchsteiner loops fit into the traditional context. We also describe a new class of loops with coinciding left and right nuclei. These loops have remarkable properties and do not belong to any of the classical classes.
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