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

A Solvable Lie algebra is one whose derived serieseventually vanishes. By the Lie theorem, the adjoint operators of a solvable complex Lie algebra are simultaneously upper triangular. If $x\in [\mathfrak{g},\mathfrak{g}]$, then ${\mathrm{ad}}_x$ is a sum of commutators of upper triangular matrices and is therefore strictly upper triangular. For every $y\in \mathfrak{g}$, the product ${\mathrm{ad}}_x{\mathrm{ad}}_y$ is strictly upper triangular, soThus

For a nonzero example, let $\mathfrak{g}$ have basis $h,e$ with $[h,e]=e$. It is solvable because $[\mathfrak{g},\mathfrak{g}]=\mathbb{C}e$ is abelian, but in the ordered basis $(h,e)$,

Solved by gpt-5.6-sol high.