Engel’s Theorem
When we say that a Lie algebra 
is nilpotent, another way of putting it is that for any sufficiently long sequence 
of elements of the nested adjoint
![\mathrm{ad}(x_n)\left[\dots\mathrm{ad}(x_2)\left[\mathrm{ad}(x_1)[y]\right]\right]](https://s-ssl.wordpress.com/latex.php?latex=%5Cmathrm%7Bad%7D%28x_n%29%5Cleft%5B%5Cdots%5Cmathrm%7Bad%7D%28x_2%29%5Cleft%5B%5Cmathrm%7Bad%7D%28x_1%29%5By%5D%5Cright%5D%5Cright%5D&bg=e6e6e6&fg=333333&s=0 "\mathrm{ad}(x_n)\left[\dots\mathrm{ad}(x_2)\left[\mathrm{ad}(x_1)[y]\right]\right]")
is zero for all 
. In particular, applying 
enough times will eventually kill any element of . That is, each 
is ad-nilpotent. It turns out that the converse is also true, which is the content of Engel’s theorem.
But first we prove this lemma: if 
is a linear Lie algebra on a finite-dimensional, nonzero vector space 
that consists of nilpotent endomorphisms, then there is some nonzero 
for which 
for all 
.
If 
then is spanned by a single nilpotent endomorphism, which has only the eigenvalue zero, and must have an eigenvector 
, proving the lemma in this case.
If 
is any nontrivial subalgebra of then 
is nilpotent for all 
. We also get an everywhere-nilpotent action on the quotient vector space 
. But since 
, the induction hypothesis gives us a nonzero vector 
that gets killed by every . But this means that ![[k,x]\in K](https://s-ssl.wordpress.com/latex.php?latex=%5Bk%2Cx%5D%5Cin+K&bg=e6e6e6&fg=333333&s=0 "[k,x]\in K")
for all , while 
. That is, is strictly contained in the normalizer 
.
Now instead of just taking any subalgebra, let be a maximal proper subalgebra in . Since is properly contained in , we must have 
, and thus is actually an ideal of . If 
then we could find an even larger subalgebra of containing , in contradiction to our assumption, so as vector spaces we can write 
for any 
.
Finally, let 
consist of those vectors killed by all 
, which the inductive hypothesis tells us is a nonempty collection. Since is an ideal, sends 
back into itself: =0](https://s-ssl.wordpress.com/latex.php?latex=k%28l%28w%29%29%3Dl%28k%28w%29%29-%5Bl%2Ck%5D%28w%29%3D0&bg=e6e6e6&fg=333333&s=0 "k(l(w))=l(k(w))-[l,k](w)=0")
. Picking a as above, its action on is nilpotent, so it must have an eigenvector 
with 
. Thus 
for all 
.
So, now, to Engel’s theorem. We take a Lie algebra consisting of ad-nilpotent elements. Thus the algebra 
consists of nilpotent endomorphisms on the vector space , and there is thus some nonzero 
for which ![[L,z]=0](https://s-ssl.wordpress.com/latex.php?latex=%5BL%2Cz%5D%3D0&bg=e6e6e6&fg=333333&s=0 "[L,z]=0")
. That is, has a nontrivial center — 
.
The quotient 
thus has a lower dimension than , and it also consists of ad-nilpotent elements. By induction on the dimension of we assume that is actually nilpotent, which proves that itself is nilpotent.
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August 22, 2012 -
Posted by John Armstrong |
Algebra, Lie Algebras
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