Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 19J / d / Solution

Take $t\mapsto A$ and $r\mapsto B$. Then $A^2=I$ andso these matrices define a representation. The line $\mathbb{C}{e}_1$ is fixed by both matrices and is a one-dimensional subrepresentation.

Any complementary line has a generator $e_2+c{e}_1$. Butwhich cannot lie on the same line. Thus no complement is $B$-invariant, and therefore no $G$-invariant complement exists.