  We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is
weakly amenable only if $G$ is abelian. Our main new idea is to show that weak
amenability of $A(G)$ implies that the anti-diagonal,
$\check{\Delta}_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for
$A(G\times G)$. We then show that this cannot happen if $G$ is non-abelian. We
conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable
only if it contains no closed connected non-abelian Lie subgroups. In
particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its
connected component of the identity $G_e$ is abelian.
