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, , is a set of local synthesis for A(G×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 Ge is abelian.