Suppose we have a symmetric (0,2) tensor $k$. (The original setting is that $k$ is the scalar second fundamental form for a Riemannian hypersurface, but I don't think it matters in my specific question.)
As we know, $D_X$ commutes with $tr_g$, so we can compute as follows : $$ \sum_m (D_X k)(E_m, E_m) = tr_g (D_X k) = D_X(tr_g k) = \sum_m D_X (k(E_m, E_m)) = \sum_m (D_X k)(E_m, E_m) + 2 \sum_m k(D_X E_m, E_m), $$ where the last step follows from Leibniz rule. However, this implies $$\sum_m k(D_X E_m, E_m) = 0.$$
My question is, if we do computations the other way around, say, to prove the commutativity and compute in orthonormal frame, then how to prove this sum vanishes. I feel like this is hard and I don't know how to proceed. I know that we can prove easily in local coordinates, but I'm just curious about this since this equation vanishes looks impossible at the first sight.