In the picture below, I think $L_Vg=0$ equal to $g(\nabla_VX,Y)+g(X,\nabla_VY)=0$ . But if so , I can't get equation of 2.7 . What's wrong ?
Picture below is from SCALAR CURVATURE, KILLING VECTOR FIELDS AND HARMONIC ONE-FORMS ON COMPACT RIEMANNIAN MANIFOLDS
Just choose $X,Y$ to be coordinate vector fields and you get $$g(\nabla_{\partial_i} V, \partial_j)=g(V^k{}_{,i}\partial_k,\partial_j)=V^k{}_{,i}g_{kj}=V_{j,i}.$$ Symmetrizing this gives you the correspondence you want.