Strongly polynormal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed strongly polynormal in $$G$$ if for any $$g \in G$$, the following identity holds:

$$[[,H],H] = [,H]$$

Here $$[K,L]$$ denotes the subgroup generated by commutators between elements of $$K$$ and elements of $$L$$.

Stronger properties

 * Strongly paranormal subgroup

Weaker properties

 * Polynormal subgroup