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