2-local Baer correspondence

Definition

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:

Baer alternating loop rings $\leftrightarrow$ Baer diassociative loops

The correspondence:

Baer Lie rings $\leftrightarrow$ Baer Lie groups

is a subcorrespondence of this correspondence.

Observations

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.