1-closed subquandle of a group

Definition
Suppose $$G$$ is a group and $$S$$ is a non-empty subset of $$G$$. We say that $$S$$ is a 1-closed subquandle of $$G$$ if the following hold:


 * 1) $$S$$ is a 1-closed subset of $$G$$, i.e., the cyclic subgroup generated by any element of $$S$$ is in $$S$$.
 * 2) For any $$a,b \in S$$, the conjugate $$aba^{-1}$$ is in $$S$$. Note that by the preceding, this is equivalent to requiring that for any $$a,b \in S$$, the conjugate $$a^{-1}ba$$ is in $$S$$. In particular, $$S$$ is a subquandle of the quandle given by the conjugation rack of $$G$$.