Converse geodesic characterization of a Killing vector ($\frac{d}{ds}(K_a{V}^a)=0\text{ for all timelike geodesics}⟺{\nabla }_{(a}{K}_{b)}=0$)

Along an affinely parametrized geodesic, differentiating the contraction of a vector field $K$ with its tangent $V$ gives $d(K_a{V}^a)/ds=V^a{V}^b{\nabla }_{(a}{K}_{b)}$. If this is zero along every timelike geodesic, arbitrary timelike initial tangents at each point make the symmetric quadratic form vanish on the whole timelike cone. That cone is open, so its quadratic polynomial must vanish identically. Thus $K$ is a Killing vector. The converse follows directly from the same identity. One selected orbit cannot establish the converse; the quantifier over all initial timelike directions is essential.