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As a novice to Lie algebras, I have been presented with Engel's Theorem quite soon. It states the following: Let $L$ be a Lie algebra of dimension $n$ over $k$ such that for every $x,y$, the “bracket chain” $$ [x[x\ldots[xy]\ldots]] $$ eventually vanishes. Then for some $d\in \mathbb N$, every chain of the more general form $$ [x_d[x_{d-1}\ldots[x_1y]\ldots]] $$ must vanish as well. This can of course be summed up by saying that “ad-nilpotency implies nilpotency”.

Most of the proofs rely on the fact that every centerless lie algebra admits an injective representation as a matrix lie algebra, i.e. a subalgebra of $\mathfrak{gl} (n)$. It is then proved inductively over $n$, i.e. the dimension of the representation.

But is there a “direct” proof which only makes use of the abstract properties of the commutator, and does not rely on a representation as a matrix algebra or the commutator lie algebra of an enveloping associative algebra (the latter of which is used extensively by Jacobson in his proof)?

Lukas Juhrich
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  • The standard proof (see Humphreys) only uses the adjoint representation and reduces the question to vector spaces and induction, yes. But this is the easiest way, I think. No need to use enveloping algebras. For me, the adjoint representation is direct, i.e., only relies on commutators because $\operatorname{ad}(x)(y)=[x,y]$. Also, matrices and vctor spaces are "direct", I think. – Dietrich Burde Jun 08 '20 at 11:00

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