Finite upper central series members are purely definable

Statement
Suppose $$G$$ is a group. Let $$Z^k(G)$$ be the (finite) fact about::upper central series of $$G$$. Explicitly, $$Z^k(G)$$ for all nonnegative integers $$k$$ is defined as follows:


 * $$Z^0(G)$$ is the trivial subgroup
 * $$Z^k(G)$$ is the subgroup containing $$Z^{k-1}(G)$$ such that $$Z^k(G)/Z^{k-1}(G)$$ is the center of $$G/Z^{k-1}(G)$$.

Then, $$Z^k(G)$$ is a purely definable subgroup of $$G$$ for each $$k$$.

Facts used

 * 1) uses::Center is purely definable
 * 2) uses::Pure definability is quotient-transitive

Direct proof
We have $$g \in Z^k(G)$$ if it satisfies the following sentence $$\varphi(g)$$:

$$\varphi(g) = \ \forall \ x_1, x_2, \dots, x_k \in G, \ : \ [[\dots [g,x_1],x_2], \dots, x_k] = e$$

where $$e$$ is the identity element.

Proof using given facts
The proof follows by induction using facts (1) and (2).