Non-Abelian Extension Theory for Lie Rings

In this paper, we establish fundamental results concerning non-abelian extensions of Lie rings. Specifically, we demonstrate one-to-one correspondence between the equivalence classes of non-abelian extensions of Lie ring [Formula: see text] by Lie ring [Formula: see text] and the elements of second cohomology group [Formula: see text], where [Formula: see text] acts on [Formula: see text]. Additionally, we provide necessary and sufficient condition under which such extensions exist, particularly when a specific abstract kernel is involved.