I'm trying to show that the sum of two killing vector fields is again a vector field without introducing a connection and just using the smooth structure on a manifold.
Let $X,Y$ be Killing vector fields so $L_X g = L_Y g = 0$.
My attempt at this so far has been to take integral curves $\phi$ and $\psi$ for $X$ and $Y$ and show that to first order in $t$ $\phi+\psi$ is the integral curve for $X+Y$ and then the result would follow from the definition of the Lie derivative as the derivative of the metric pullback by the integral curve.
I'm not sure if this is the right way to proceed and I haven't been able to show this so would appreciate any help.