All minimal CL-subgroups have the same derived subgroup

Statement
Suppose $$P$$ is a group of prime power order. Then, if $$A,B$$ are two fact about::minimal CL-subgroups of $$P$$, the fact about::commutator subgroups $$[A,A]$$ and $$[B,B]$$ are equal.

Textbook references

 * Corollary 4.2 (Page 490) of