\begin{document}$a_1(1420)$\end{document} observed by the COMPASS Collaboration has been widely accepted. In this work, we investigate the triangle mechanism in the decay \begin{document}$B^0 \to D^- \pi^+ a_0(980)(\pi^0 \eta)$\end{document}, where the \begin{document}$a_0(980)$\end{document} is treated as a dynamically generated state. The \begin{document}$\bar{K}^{*0}$\end{document}-\begin{document}$K^+$\end{document}-\begin{document}$K^-$\end{document} triangle loop originates from the decay \begin{document}$B^0 \to D^- K^+ \bar{K}^{*0}$\end{document}, which has been observed by the Belle Collaboration, followed by the subsequent decay \begin{document}$\bar{K}^{*0} \to K^- \pi^+$\end{document}. The triangle amplitude develops a pronounced peak around 1420 MeV, which is reflected in the invariant mass spectrum of the \begin{document}$\pi a_0(980)$\end{document} system. The differential decay width is calculated and exhibits a narrow peak around \begin{document}$980~\mathrm{MeV}$\end{document} in the \begin{document}$\pi^0 \eta$\end{document} invariant mass distribution. Furthermore, the invariant mass distribution of the \begin{document}$\pi^+ \pi^0 \eta$\end{document} system shows a clear peak around \begin{document}$1420~\mathrm{MeV}$\end{document}, which further confirms the triangle singularity explanation of the \begin{document}$a_1(1420)$\end{document}. We expect that the proposed \begin{document}$B^0$\end{document} decay mode could provide a potential platform for further exploring the triangle-singularity nature of the \begin{document}$a_1(1420)$\end{document} and could be tested in future experiments such as LHCb, BESIII, and Belle II."> The role of triangle singularity in the <inline-formula><tex-math id="Z-20260731084336">\begin{document}$ {B^0 \to D^- \pi^+ a_0(980)(\pi^0 \eta)}$\end{document}</tex-math></inline-formula> decay -
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