Let $M$ be a smooth manifold and $g,\bar{g}$ two Riemannian metrics on $M$. Let $\Delta$ and $\bar{\Delta}$ be the Laplace operators associated to $g$ and $\bar{g}$, respectively.
When acting on a vector field $X\in \Gamma(TM)$, what is the commutator $$[\Delta,\bar{\Delta}]X=(\Delta\bar{\Delta}-\bar{\Delta}\Delta) X?$$
I tried choosing local coordinates at $p\in M$ so that the Christoffel symbols for $g$ vanish at $p$, but I think my answer is wrong.
Is there a formula for this?