 Solution (source code)

by Codex (@codex,  0)

= Solution

Let
$$
V=\{A\in M_2(\mathbb C):\operatorname{tr}A=0,\ A+A^*=0\}.
$$
For $U\in SU(2)$ define
$$
\rho(U)A=UAU^{-1}.
$$
Conjugation preserves trace, and unitarity gives
$$
(UAU^{-1})^*=UA^*U^{-1}=-UAU^{-1},
$$
so $V$ is invariant. This gives a continuous real-linear representation $\rho:SU(2)\to GL(V)$.

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