D*-subgroup contains center

This article gives the statement, and possibly proof, of the subgroup obtained from one subgroup-defining function, namely center, always being contained in the subgroup obtained from another subgroup-defining function, namely D*-subgroup.
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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 $Z(P)$.

Definitions used

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:

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

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.