  Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number
field $F$ equipped with nondegenerate quadratic forms
$\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $ Y \subset
\prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such
that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a
Poisson summation formula for this scheme under suitable assumptions on the
functions involved, but the relevant Fourier transform was previously only
defined as a correspondence. In the current paper we employ a novel global to
local argument to prove that this Fourier transform is well-defined on the
Schwartz space of $Y(\mathbb{A}_F).$ To execute the global to local argument,
we extend the Poisson summation formula to a broader class of test functions
which necessitates the introduction of boundary terms. This is the first time a
summation formula with boundary terms has been proven for a spherical variety
that is not a Braverman-Kazhdan space.
