I saw this come up in a lecture on conformal field theory, and was a bit skeptical of the claim. So, given some conformal Killing field,
$$ \mathcal{L}_{\xi}~g = \lambda g $$ it was said the following is true $$ \mathcal{L}_{\nabla^2\xi}g=g\nabla^2\lambda $$ Where $\nabla^2=\nabla^a\nabla_a$ (a contraction of covariant derivatives). I have tried writing things out but can't get anywhere. I end up with a bunch of Riemann tensor terms that don't cancel. So, first off is this even true, and if so how can I show it?