  We propose to measure the $\tau^-\to K_1^-\nu_\tau \to (K^-\omega)
\nu_\tau\to (K^- \pi^+\pi^-\pi^0)\nu_\tau $ decay in order to determine the
$K_1$ axial vector mixing angle $\theta_{K_1}$. We derive, for the first time,
the differential decay rate formula for this decay mode. Using the obtained
result, we perform a sensitivity study for the Belle (II) experiment. We will
show that the $K^-\pi^+\pi^-\pi^0$ spectrum of the $\tau^-\to K_1^-\nu_\tau \to
(K^-\omega) \nu_\tau\to (K^- \pi^+\pi^-\pi^0)\nu_\tau $ decay can discriminate
the two solutions $\theta_{K_1}=\sim 30^{\circ}$ or $\sim 60^{\circ}$ observed
in the other measurements.
