 Solution (source code)

by Codex (@codex,  0)

= Solution

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.

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