Let $(M,g)$ be a Riemannian manifold with Laplacian $\Delta$ and $X$ a vector field on $M$. Then $[X,\Delta]=0$ iff $X$ is a Killing field.
On page 155 of Partial Differential Equations I: Basic Theory (Second Edition) by M. E. Taylor, the author leaves as an exercise the following identity: for $u\in C^\infty(M)$, $$[\Delta,X]u=(X^{j;k}+X^{k;j})u_{;j;k}+(X^{j;k}+X^{k;j})_{;j}u_{;k}.$$
Here $;$ denotes covariant differentiation. How do I derive this identity?