I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-dependent proof but my rationale seems to be rather different from the coordinate-dependent proof.
Be $K$ a Killing vector and $\gamma$ the geodesic and $\dot{\gamma}$ its tangent vector. $g(\cdot, \cdot)$ is the scalar product based on the metric $g$. $\nabla_V X$ is the covariant derivative.
$$d g(K,\dot{\gamma})(\dot{\gamma}) = \nabla_\dot{\gamma}g( K,\dot{\gamma}) = g( \nabla_\dot{\gamma}K, \dot{\gamma}) + g( K, \nabla_\dot{\gamma}\dot{\gamma}) = g( \nabla_\dot{\gamma}K, \dot{\gamma}) $$
where the second term vanishes because of $\nabla_\dot{\gamma}\dot{\gamma}=0$ along the geodesic. The first term also vanishes since I can conclude from the coordinate-independent form of the Killing vector equation:
$$g(\nabla_V K,W) + g(V,\nabla_W K)=0$$
that
$$g(\nabla_\dot{\gamma}K,\dot{\gamma})=0$$
So $$d g(K,\dot{\gamma})(\dot{\gamma})=0$$
which shows the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic. Is this derivation correct?