Definition
A Lie algebra is called nilpotent if the lower central series terminates.
That is, if there exists such that .
Properties
Preserved under sub and quotient object
Let be a nilpotent Lie algebra, with a nilpotent and an ideal .
The Lie algebras and are nilpotent.
Proof
Note for a subalgebra then .
Therefore, the lower central series will terminate for both Lie algebras.
Let be a surjective Lie algebra homomorphism.
We have .
Then is surjective by induction since
Therefore, if the series terminates for then by the homomorphism it will for the quotient as well.