Center of a multiplicative Lie ring

Definition
Suppose $$L$$ is a multiplicative Lie ring. The center of $$L$$, denoted $$Z(L)$$, is defined as the subset:

$$\{ x \in L \mid xy = yx \mbox{ and } \{ x,y \} = 1 \ \forall \ y \in L \}$$

Here, $$1$$ denotes the identity element of the underlying group of $$L$$.

The center of a multiplicative Lie ring is an ideal, and in particular, it is a subring, of the multiplicative Lie ring.