  The classical Shimura correspondence lifts automorphic representations on the
double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take
key steps towards establishing a relative trace formula that would give a new
global Shimura lift, from the triple cover of $SL_3$ to $PGL_3$, and also
characterize the image of the lift. The characterization would be through the
nonvanishing of a certain global period involving a function in the space of
the automorphic minimal representation $\Theta_{SO_8}$ for split
$SO_8({\mathbb{A}})$, consistent with a 2001 conjecture of Bump, Friedberg and
Ginzburg. In this paper, we first analyze a global distribution on
$PGL_3({\mathbb{A}})$ involving this period and show that it is a sum of
factorizable orbital integrals. The same is true for the Kuznetsov distribution
attached to the triple cover of $SL_3({\mathbb{A}})$. We then match the
corresponding local orbital integrals for the unit elements of the spherical
Hecke algebras; that is, we establish the Fundamental Lemma.
