  Schramm-Loewner evolution (SLE) is a random process that gives a useful
description of fractal curves. After its introduction, many works concerning
the connection between SLE and conformal field theory (CFT) have been carried
out. In this paper, we develop a new method of coupling SLE with a
Wess-Zumino-Witten (WZW) model for $SU(2)$, an example of CFT, relying on a
coset construction of Virasoro minimal models. Generalizations of SLE that
correspond to WZW models were proposed by previous works [Bettelheim {\it et
al.}, Phys. Rev. Lett. {\bf 95}, 251601 (2005)] and [Alekseev {\it et al.},
Lett. Math. Phys. {\bf 97}, 243-261 (2011)], in which the parameters in the
generalized SLE for $SU(2)$ were related to the level of the corresponding
$SU(2)$-WZW model. The present work unveils the mechanism of how the parameters
were chosen, and gives a simpler proof of the result in these previous works,
shedding light on a new perspective of SLE/WZW coupling.
