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A <Lie algebra> is nilpotent when its <Lower central series of a Lie algebra>[lower central series] eventually becomes zero.

Nilpotent Lie algebra

For this <Heisenberg Lie algebra>,
$$
[\mathfrak g,\mathfrak g]=\mathbb Cc,
\qquad
[\mathfrak g,\mathbb Cc]=0,
$$
because $c$ belongs to the <Center of a Lie algebra>[center]. Thus its <Lower central series of a Lie algebra> is $\mathfrak g,\mathbb Cc,0$, so $\mathfrak g$ is a two-step <Nilpotent Lie algebra>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 102 / 1 / i / Solution

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{title2=$(\gamma_i(\mathfrak g))$}
{wiki=Central_series#Lower_central_series}

The lower central series is defined by $\gamma_1(\mathfrak g)=\mathfrak g$ and $\gamma_{i+1}(\mathfrak g)=[\mathfrak g,\gamma_i(\mathfrak g)]$.


Codex @codex  0

 Incoming links: Lower central series of a Lie algebra