  The role of triangle mechanism in the decay process $J/\psi \to K^- K^+
a_1(1260)$ is probed. In this mechanism, a close-up resonance with mass $1823$
MeV and width $122$ MeV decays into $K^* \phi, K^* \to K \pi$ and then $K^*
\bar{K}$ fuses into the $a_1(1260)$ resonance. We find that this mechanism
leads to a triangle singularity around $M_{\rm inv}(K^- a_1(1260))\approx 1920$
MeV, where the axial-vector meson $a_1(1260)$ is considered as a dynamically
generated resonance. With the help of the triangle mechanism we find sizable
branching ratios $\text{Br}(J/\psi \to K^- K^+ a_1(1260),a_1 \to \pi
\rho)=1.210 \times 10^{-5}$ and $\text{Br}(J/\psi \to K^- K^+ a_1(1260))=3.501
\times 10^{-5}$. Such a effect from triangle mechanism of the decay process
could be investigated by such as BESIII, LHCb and Belle-II experiments. This
potential investigation can help us obtain the information of the axial-vector
meson $a_1(1260)$.
