= Solution
Realize the generators as affine reflections of $\mathbb R^2$:
$$
s(x,y)=(-x,y),
\qquad
u(x,y)=(x,-y),
\qquad
t(x,y)=(1-y,1-x).
$$
The reflecting lines for $s$ and $u$ are perpendicular, while the line $x+y=1$ meets each at angle $\pi/4$. Hence
$$
s^2=t^2=u^2=(su)^2=(st)^4=(tu)^4=1,
$$
so the presentation maps onto this affine reflection group. But
$$
g=sut,qquad g(x,y)=(y-1,x-1),qquad
g^2(x,y)=(x-2,y-2),
$$
and $g$ has infinite order. Thus $W$ is infinite. The finite Coxeter group $W(B_4)$ is the <signed symmetric group> on four letters and has order $2^4 4!$. An infinite group cannot embed in it, so $W$ is not a subgroup of $W(B_4)$.
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