Related to Why symplectic geometry gives Poisson geometry

Is there any common use for the Lie bracket of the gradients of two functions? That is, $[\nabla f, \nabla g]$?

Here, $\nabla f$ denotes the vector field gotten by "raising the index" from $df$. In other words, $(\nabla f)(g) = m(df, dg)$, where $m$ is the Riemannian metric on some manifold.