Homoclinism of Lie rings

Short definition
A homoclinism of Lie rings is a homologism of Lie rings with respect to the subvariety of the variety of Lie rings given by abelian Lie rings.

Full definition
For any Lie ring $$L$$, let $$Z(L)$$ denote the center of $$L$$, $$\operatorname{Inn}(L)$$ denote the Lie ring of inner derivations of $$L$$ (explicitly, it is isomorphic to $$L/Z(L)$$, and $$L'$$ denote the derived subring of $$L$$.

Let $$\gamma_L$$ denote the mapping $$\operatorname{Inn}(L) \times \operatorname{Inn}(L) \to L'$$ that arises from the Lie bracket mapping $$L \times L \to L'$$, and then observing that this map is constant on the cosets of $$Z(L) \times Z(L)$$. Note that the mapping is $$\mathbb{Z}$$-bilinear.

A homoclinism of Lie rings $$L$$ and $$M$$ is a pair $$(\zeta,\varphi)$$ where $$\zeta$$ is a homomorphism of $$\operatorname{Inn}(L)$$ with <math\operatorname{Inn}(M) and $$\varphi$$ is a homomorphism of $$L'$$ with $$M'$$, such that $$\varphi \circ \gamma_L = \gamma_M \circ (\zeta \times \zeta)$$.