Inner derivation implies endomorphism for class two Lie ring

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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.
View other analogues of class two implies commutator map is endomorphism|View other analogues from group to Lie ring (OR, View as a tabulated list)


Suppose L is a Lie ring of nilpotency class two (?). In other words, [[x,y],z] = 0 for all x,y,z \in L. Then, any inner derivation of L is an endomorphism of L as a Lie ring.

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Opposite facts

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Given: A Lie ring L of nilpotency class two, an element x \in L.

To prove: The map y \mapsto [x,y] is an endomorphism of L as a Lie ring.

Proof: The map is clearly an endomorphism of the additive group of L, so it suffices to show that it preserves the Lie bracket. In other words, we need to show that for y, z \in L, we have:

\! [x,[y,z]] = [[x,y],[x,z]].

But since L has nilpotency class two, both sides are zero, so the result holds.