  We consider the parabolically induced representations of the symmetric space
$SO_4\backslash G_2$ over a p-adic field using the geometric lemma when the
inducing parabolic is $P_{\beta}$. Using an explicit description of the
embedding of $G_2$ in $GL_8$, we characterize precisely the induced
representations which are $(SO_4, \chi)$-distinguished, given a certain type of
involutions is chosen.
