Classification of Lie rings of prime-square order

Statement
Let $$p$$ be a prime number. There are only three possibilities, up to isomorphism, for a Lie ring of prime-square order $$p^2$$:


 * 1) Abelian Lie ring whose additive group is cyclic group of prime-square order.
 * 2) Abelian Lie ring whose additive group is elementary abelian group of prime-square order.
 * 3) Nontrivial semidirect product of Lie rings of prime order.

Facts used

 * 1) uses::Structure theorem for finitely generated abelian groups

First part: identifying the possible additive groups
By Fact (1), the possibilities for the additive group, namely, the abelian groups of order $$p^2$$, correspond precisely to the possible partitions of the number $$2$$:

Second part: dealing with the case of the cyclic group of prime-square order
If the additive group of a Lie ring is cyclic, then the Lie ring must be abelian, because the Lie bracket, being alternating, must vanish on any cyclic subgroup. Thus, for the case where the additive group of cyclic of prime-square order, the only possibility is that of an abelian Lie ring.

Third part: dealing with the case of the elementary abelian group of prime-square order
One possibility is that we have an abelian Lie ring. Let's consider the case that the Lie ring is non-abelian and try to deduce its structure.