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Let $Q$ be a loop. If $S\leq Q$ is such that $\varphi(S) \subseteq S$ foreach standard generator of $\mathrm{Inn}(Q)$, then $S$ does not have to be anormal subloop. In an LC loop the left and middle nucleus coincide and form anormal subloop. The identities of Osborn loops are obtained by applying theidea of nuclear identification, and various connections of Osborn loops toMoufang and CC loops are discussed. Every Osborn loop possesses a normalnucleus, and this nucleus coincides with the left, the right and the middlenucleus. Loops that are both Buchsteiner and Osborn are characterized as loopsin which each square is in the nucleus.

Normality, nuclear squares and Osborn identities