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In mathematics, a Lie algebra is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras

We write , and for all If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions. Note that the definition means that, viewed as a non associative non unital algebra, a Lie algebra is nilpotent if it is nilpotent as an ideal.

If is the set of k × k matrices with entries in , then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra.

A Heisenberg algebra is nilpotent. For example, in dimension 3, the commutator of two matriceswhere .

A Cartan subalgebra of a Lie algebra is nilpotent and self normalizing page 80. The self normalizing condition is equivalent to being the normalizer of a Lie algebra. This means This includes upper triangular matrices and all diagonal matrices in .

If a Lie algebra has an automorphism of prime period with no fixed points except at 0, then is nilpotent.

Every nilpotent Lie algebra is solvable. This is useful in proving the solvability of a Lie algebra since, in practice, it is usually easier to prove nilpotency (when it holds!) rather than solvability. However, in general, the converse of this property is false. For example, the subalgebra of (k ≥ 2) consisting of upper triangular matrices, , is solvable but not nilpotent.

If a Lie algebra is nilpotent, then all subalgebras and homomorphic images are nilpotent.

If the quotient algebra , where is the center of , is nilpotent, then so is This is to say that a central extension of a nilpotent Lie algebra by a nilpotent Lie algebra is nilpotent.

Engel's theorem: A finite dimensional Lie algebra is nilpotent if and only if all elements of are ad nilpotent.

The Killing form of a nilpotent Lie algebra is 0.

A nonzero nilpotent Lie algebra has an outer automorphism, that is, an automorphism that is not in the image of Ad.

The derived subalgebra of a finite dimensional solvable Lie algebra over a field of characteristic 0 is nilpotent.