Requirements in Engel's theorem

Question

I have learned the statement of Engel's theorem. The condition on $L$ is that it must be finite dimensional.

Now I can ask the question: Why? Can someone give me an example of an infinite dimensional $L$, where Engel's theorem does not apply?

Answer 1

Here's a concrete example with matrices.

Take the Lie algebra of all strictly upper triangular infinite matrices $M=(m_{ij})_{i,j \in \mathbf{Z}_{>0}}$ with rows and columns indexed by the positive integers and with only finitely many non-zero entries $m_{ij}$ in some ring (thanks to the finiteness condition, you can multiply these as usual, and hence the commutator is well-defined). This is not nilpotent (its central series may be described in the same way as for finite strictly upper triangular matrices!), but evidently each element is nilpotent and hence ad-nilpotent.

The philosophy that local nilpotent implies global nilpotence fails here just because local and global are so far apart for big Lie algebras.