  Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier
algebra of $G$. We show that the anti-diagonal
$\check{\Delta}_G=\{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of
local smooth synthesis and a set of local weak synthesis of degree at most
$[\frac{n}{2}]+1$ for $A(G\times G)$. We achieve this by using the concept of
the cone property in \cite{ludwig-turowska}. For compact $G$, we give an
alternative approach to demonstrate the preceding results by applying the ideas
developed in \cite{forrest-samei-spronk}. We also present similar results for
sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times
G\times G$ of diagonal forms. Our results very much depend on both the
geometric and the algebraic structure of these sets.
