CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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A nilpotent-by-nilpotent extension need not be nilpotent

Counterexample

The assertion that an extension of a nilpotent Lie algebra by a nilpotent Lie algebra must be nilpotent is false. For the affine algebra $a = k x \oplus k y$ with $[x, y] = y$ , the sequence

$0 ⟶ k y ⟶ a ⟶ a / k y ⟶ 0$

has nilpotent kernel and quotient, but $a$ is not nilpotent.

Facts & Assumptions

Given: The displayed affine Lie algebra over a field $k$ , with the first map the inclusion and the second the quotient map.

[L1]

Its lower central series satisfies $γ_{r} (a) = k y \neq 0$ for every $r \geq 2$ (The two-dimensional affine Lie algebra is solvable, not nilpotent).

[L2]

The quotient bracket is $[u + I, v + I] = [u, v] + I$ for an ideal $I$ (Quotient Lie algebras).

[L3]

Every nonzero abelian Lie algebra is nilpotent of class one (Abelian Lie algebras are nilpotent of class one).

Refutation

technique · counterexample

1.1

The line $k y$ is an ideal because $[x, y] = y$ and $[y, y] = 0$ . The displayed inclusion is injective, the quotient map is surjective, and its kernel is exactly $k y$ , so the sequence is short exact.

givenalgebra

1.2

Nevertheless, [L1] says that every lower-central term of $a$ from $γ_{2}$ onward is the nonzero line $k y$ . Thus the ambient algebra is not nilpotent.

2.1

The kernel $k y$ is one-dimensional abelian. The quotient is spanned by $x + k y$ , and [L2] gives $[x + k y, x + k y] = 0 + k y$ , so it too is one-dimensional abelian. Hence both kernel and quotient are nilpotent by [L3].

L2L3step 1.1

3.1

Steps 1.1, 1.2, and 2.1 satisfy the hypotheses of a nilpotent-by-nilpotent extension and explicitly fail its proposed nilpotence conclusion. The construction is finite and uses no choice principle.

step 1.1step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Milne, Lie Algebras, nilpotent extensions (standard reference, not scraped)