2-local Baer correspondence
The 2-local Baer correspondence is a slight variation on the usual Baer correspondence that aims to establish a correspondence between things that 2-locally look like Baer Lie rings and things that 2-locally look like Baer Lie groups. Explicitly, it is a correspondence:
is a subcorrespondence of this correspondence.
The first thing we observe is that under this correspondence, it is possible for Lie rings that are not of class two (but have 2-local class two) to get mapped to diassociative loops that aren't groups. Conversely, it is possible for groups that are not of class two (but have 2-local class two) to get mapped to alternating rings that are not Lie rings.