Join-transitively subnormal subgroup

Symbol-free definition
A subgroup of a group is termed join-transitively subnormal if its join (viz., the subgroup generated) with any defining ingredient::subnormal subgroup is again subnormal.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed join-transitively subnormal if whenever $$K \triangleleft \triangleleft G$$ (viz., $$K$$ is subnormal in $$G$$), the join of subgroups $$\langle H,K \rangle$$ is subnormal in $$G$$.

Formalisms
The subgroup property of being join-transitively subnormal is obtained by applying the join-transiter to the subgroup property of being subnormal.

Metaproperties
Clearly, the whole group is join-transitively subnormal, because its join with any subgroup is the whole group. Also, the trivial subgroup is join-transitively subnormal, because its join with any subnormal subgroup is the same subnormal subgroup.

By the general theory of transiters, the join-transiter of any subgroup property is itself a finite-join-closed subgroup property.