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Every nilpotent Lie algebra is solvable (derived series terminates).
Let be a finite-dimensional Lie algebra.
If is nilpotent (defined via the lower central series), then is solvable (defined via the derived series).
This inclusion explains why nilpotent Lie algebras sit strictly inside solvable ones; the reverse implication fails in general, and distinguishing solvable from nilpotent is often done by comparing the derived and lower central filtrations.
Let be a finite-dimensional Lie algebra.
The lower central series is defined by and (see lower central series).
The Lie algebra is nilpotent if there exists such that
The smallest such is the nilpotency class (or step).
Let be a Lie algebra.
The lower central series of is the descending sequence of ideals
Each is an ideal because it is generated by brackets with .
Let be a Lie algebra. Define its derived series (see derived series) by
where denotes the Lie bracket and each derived algebra is an ideal (compare derived subalgebra and it is an ideal).
The Lie algebra is solvable if there exists such that
Equivalently, iterated commutators eventually vanish. This notion is central in the structure theory of general Lie algebras: the maximal solvable ideal is the radical, and the Levi decomposition splits any finite-dimensional Lie algebra (in characteristic ) into a semisimple part and a solvable part.
There are several important tests and relations:
Let be a Lie algebra.