Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 8F / ii

Statement (ii) is false. The condition is necessary for an idempotent, but it is not sufficient. For example,satisfies $(I-\alpha {)}^2=0$, hence $\alpha (I-\alpha {)}^2=0$, whileover, for example, $\mathbb{Q}$.