  We present a systematic study of the productions and decays of light axial
vector mesons with $J^{PC}=1^{+\pm}$ in charmonium decays. In the quark model
scenario, the two axial vector nonets are connected with each other by the
Gell-Mann-Okubo mass relation through the mixing between two $K_1$ states (i.e.
$K_1(1270)$ and $K_1(1400)$) with configurations of $^3P_1$ and $^1P_1$. The
mixing angles between $f_1$ and $f_1'$ (i.e. $f_1(1285)$ and $f_1(1420)$), and
between $h_1$ and $h_1'$ (i.e. $h_1(1170)$ and $h_1(1415)$) can be reliably
constrained. We then introduce the intermediate $K^*\bar{K}+c.c.$ meson loop
transitions in the description of the productions and decays of these axial
vector mesons. The presence of the nearby $S$-wave $K^*\bar{K}+c.c.$ to which
these axial vector mesons have strong coupling strengths, turns out to be
crucial for understanding many puzzling questions related to their productions
and decays. This is because that the $S$-wave $K^*\bar{K}+c.c.$ rescatterings
by the kaon exchange satisfy the triangle singularity (TS) condition in some of
these cases and the TS mechanism can introduce special interference effects in
the exclusive decays of these light axial vector mesons.
