Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / d / ii / Solution

The defining two-dimensional group representation is reducible, because the line $\mathbb{R}{e}_1$ is an invariant subspace:For $b\ne 0$, the complementary line $\mathbb{R}{e}_2$ is not invariant, so this representation need not split as a direct sum of one-dimensional representations.