Isoclinism of Lie rings

Short definition
An isoclinism of Lie rings is an isologism 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.

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

Two Lie rings are termed isoclinic Lie rings if there is an isoclinism of Lie rings between them.

Definition in terms of homoclinism
An isoclinism of Lie rings is a homoclinism of Lie rings that is invertible, i.e., a homoclinism for which both the component homomorphisms are isomorphisms.