  We analyse the radiative decay $\eta \to \pi^+\pi^- \pi^0\gamma$ in the
low--energy expansion of the Standard Model. We employ the notion of
``generalized bremsstrahlung'' to take full advantage of the theoretical and
experimental information on the corresponding non-radiative $\eta \to 3 \pi$
decay. The direct emission amplitude of ${\cal O}(p^4)$ is due to one-loop
diagrams with intermediate pions (isospin violating) and kaons (isospin
conserving). The isospin conserving contributions to direct emission, including
vector meson exchange appearing at ${\cal O}(p^6)$, are suppressed.
