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Groupprops β

Inner derivation implies endomorphism for class two Lie ring

Statement

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.

Related facts

Proof

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.