Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 1 / b / iv / Solution

The line $\mathbb{R}S$ is a nonzero central ideal, so the Heisenberg Lie algebra is not simple. It is two-step nilpotent and hence solvable; a nonzero solvable Lie algebra is not semisimple. It is therefore neither simple nor semisimple.