D*-subgroup contains center

Statement
Suppose $$p$$ is a prime number and $$P$$ is a finite p-group. The D*-subgroup of $$P$$, denoted $$D^*(P)$$, contains the  center of $$P$$, denoted $$Z(P)$$.

D*-subgroup
Let $$p$$ be a prime number and $$P$$ be a finite p-group. The $$D^*$$-subgroup of $$P$$, denoted $$D^*(P)$$, is defined as the unique maximal element in the collection $$\mathcal{D}^*(P)$$ of subgroups of $$P$$ defined as:

$$\mathcal{D}^*(P) = \{ A \le P \mid A \mbox{ is abelian and }\operatorname{class}(\langle A,x \rangle ) \le 2 \implies x \in C_P(A) \ \forall \ x \in P \}$$

Center
The center is the set of all elements that commute with all elements of the group.

Proof
The proof follows directly from the observation that $$Z(P) \in \mathcal{D}^*(P)$$, which is immediate from the definition.