Homomorphism of Lie rings

Definition

Suppose $L_{1}$ and $L_{2}$ are Lie rings. A homomorphism of Lie rings is a map $f:L_{1}\to L_{2}$ such that:

1. $f$ is a homomorphism of groups with respect to the additive group structures of $L_{1}$ and $L_{2}$.
2. $f([x,y])=[f(x),f(y)]$, i.e., $f$ commutes with the Lie bracket.

In other words, a homomorphism of Lie rings is a homomorphism in the variety of Lie rings.