  We study the isospin breaking in the reactions $\chi_{c1} \to \pi^0 \pi^+
\pi^-$ and $\chi_{c1} \to \pi^0 \pi^0 \eta$ and its relation to the $a_0(980) -
f_0(980)$ mixing, which was measured by the BESIII Collaboration. We show that
the same theoretical model previously developed to study the $\chi_{c1} \to
\eta \pi^+ \pi^-$ reaction (also measured by BESIII), and further explored in
the predictions to the $\eta_{c} \to \eta \pi^+ \pi^-$, can be successfully
employed in the present study. We assume that the $\chi_{c1}$ behaves as an
$SU(3)$ singlet to find the weight in which trios of pseudoscalars are created,
followed by the final state interaction of pairs of mesons to describe how the
$a_0(980)$ and $f_0(980)$ are dynamically generated, using the chiral unitary
approach in coupled channels. The isospin violation is introduced through the
use of different masses for the charged and neutral kaons, either in the
propagators of the pairs of mesons created in the $\chi_{c1}$ decay, or in the
propagators inside the $T$ matrix, constructed through the unitarization of the
scattering and transition amplitudes of pairs of pseudoscalar mesons. We find
that violating isospin inside the $T$ matrix makes the $\pi^0\eta \to
\pi^+\pi^-$ amplitude nonzero, which gives an important contribution and also
enhances the effect of the $K\bar{K}$ term. Also, we find that in the total
amplitude the most important effect is the isospin breaking inside the $T$
matrix, due to the constructive sum of $\pi^0\eta \to \pi^+\pi^-$ and $K\bar{K}
\to \pi^+ \pi^-$, which is essential to get a good agreement with the
experimental measurement of the mixing.
