Nilpotent Lie algebra

Engel’s theorem

Let be a finite-dimensional Lie algebra. Then is a nilpotent Lie algebra iff all elements are ^nilpotent.¹ lie

Proof

Let be a Lie algebra with all elements -nilpotent. Since is a Lie algebra of nilpotent endomorphisms, there exists some nonzero such that , i.e. the centre .

Thus has all elements -nilpotent and is of smaller dimension. We can repeat the process, and eventually it must bottom out with ; thus by ^P2 is nilpotent, and so on all the way back to .

For the converse, assume is nilpotent, say . Then , as required.

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Footnotes

1. 1972. Introduction to Lie Algebras and Representation Theory, §3.3, pp. 12–13 ↩