nilpotent Lie algebra

Some Lie algebras are almost abelian: brackets are not zero, but if you keep bracketing repeatedly the results shrink toward nothing in a finite number of steps. A nilpotent Lie algebra is one where iterated bracketing is so weak that it eventually dies out completely. The prototype is the strictly upper-triangular matrices, where multiplying enough of them together always gives zero.

Define the lower central series of L by L^1 = L and L^{k+1} = [L, L^k]; each term is an ideal containing the next. The algebra L is nilpotent if this descending chain reaches zero: L^{c+1} = 0 for some c, the smallest such c being the nilpotency class. Equivalently, every iterated bracket of length c+1 vanishes: [x_0, [x_1, [..., x_c]...]] = 0 for all choices.

Nilpotent implies solvable, but not conversely — solvability only requires the derived series to vanish, which is a weaker descent. Engel's theorem gives the operational test: L is nilpotent if and only if ad(x) is a nilpotent operator for every x in L. The center of a nonzero nilpotent Lie algebra is always nonzero, mirroring the same fact for nilpotent groups.