2-powered twisted subgroup

Definition
A subset $$S$$ of a group $$G$$ is termed a 2-powered twisted subgroup if it satisfies both the following conditions:


 * 1) For every $$x \in S$$, there exists a unique element $$y \in S$$ such that $$y^2 = x$$.
 * 2) $$S$$ is a twisted subgroup of $$G$$, i.e., it contains the identity element, is closed under taking inverses, and for every $$x,y \in S$$, we have $$xyx \in S$$.

Particular cases

 * We can restrict the twisted multiplication on a 2-powered group to any 2-powered twisted subgroup.
 * 2-powered twisted subgroups in Baer Lie groups are in correspondence with 2-powered subgroups of the additive group of the Lie ring under the Baer correspondence.