A vector subspace $I\subseteq\mathfrak g$ is an <ideal of a Lie algebra> when $[\mathfrak g,I]\subseteq I$. A <Nilpotent Lie algebra> is one whose lower central series
$$
\mathfrak g=\gamma_1(\mathfrak g),\qquad
\gamma_{r+1}(\mathfrak g)=[\mathfrak g,\gamma_r(\mathfrak g)]
$$
eventually vanishes.

By the <Engel theorem>, the operators $\operatorname{ad}_x$ for a nilpotent complex Lie algebra can be represented simultaneously by strictly upper triangular matrices. Their products are strictly upper triangular and have zero trace. Hence
$$
\boxed{\kappa(x,y)=0\quad\text{for every }x,y\in\mathfrak g}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 1 / b / Solution

{c}
{wiki=Engel's_theorem}

A finite-dimensional Lie algebra is nilpotent if and only if every adjoint map $\operatorname{ad}_x$ is nilpotent. A Lie algebra of nilpotent endomorphisms can be simultaneously represented by strictly upper triangular matrices.


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 Incoming links: Engel theorem