The defining two-dimensional <group representation> is <reducible representation>[reducible], because the line $\mathbb Re_1$ is an <invariant subspace>:
$$
\begin{pmatrix}a&b\\0&1\end{pmatrix}e_1=ae_1.
$$
For $b\ne0$, the complementary line $\mathbb Re_2$ is not invariant, so this representation need not split as a <direct sum> of one-dimensional representations.

Solution

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

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A nonzero representation is reducible when it has a nonzero proper <invariant subspace>; otherwise it is an <irreducible representation>.

Reducible representation

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An invariant subspace of a representation $V$ is a <vector subspace> $W\subseteq V$ satisfying $\rho(g)W\subseteq W$ for every represented group element $g$.

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 Incoming links: Invariant subspace