Inner derivation implies endomorphism for class two Lie ring
ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: class two implies commutator map is endomorphism.
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Suppose is a Lie ring of nilpotency class two (?). In other words, for all . Then, any inner derivation of is an endomorphism of as a Lie ring.
- Derivation equals endomorphism for Lie ring iff it is abelian
- Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant
Given: A Lie ring of nilpotency class two, an element .
To prove: The map is an endomorphism of as a Lie ring.
Proof: The map is clearly an endomorphism of the additive group of , so it suffices to show that it preserves the Lie bracket. In other words, we need to show that for , we have:
But since has nilpotency class two, both sides are zero, so the result holds.