Isoclinic Lie rings

Definition
Two Lie rings are said to be isoclinic if there is an defining ingredient::isoclinism of Lie rings between them.

Lie rings isoclinic to the trivial ring
A Lie ring is isoclinic to the trivial Lie ring if and only if it is an abelian Lie ring.

Subrings isoclinic to each other
Any subring of a Lie ring is isoclinic to its product with the center of the ring. In particular, this means that any two subrings having nonempty intersections with the same cosets of the center are isoclinic to each other. It also implies that any cocentral subring of a Lie ring is isoclinic to the whole Lie ring.

Simple invariants

 * Isoclinic Lie rings have same nilpotency class (with the minor issue of class zero versus one; in particular, any Lie ring isoclinic to a nilpotent Lie ring is nilpotent).
 * Isoclinic Lie rings have same derived length (in particular, any Lie ring isoclinic to a solvable Lie ring is solvable)
 * Isoclinic Lie rings have same non-abelian composition factors

Probabilistic invariants

 * Isoclinic Lie rings have same commuting fraction