I am wondering if there is a nice (ideally coordinate free, something that holds in manifolds) identity of the form $\nabla \times [X,Y] = [\nabla \times X,Y] + [X, \nabla \times Y] + ...$, and if this is not something that can exist is it at least possible to determine that if $X,Y$ are irrotational $\nabla \times X = \nabla \times Y = 0$ then we can conclude their Lie bracket is irrotational.
I first tried in vain writing $\nabla \times = \star d$ and hoping I could get some very easy result out of exterior derivative identities but the Lie bracket completely brick walled any progress from this direction. Since that didn't work I calculated $(\nabla \times v)^i= \varepsilon^i_{jk} \frac{\partial}{\partial x^j}v^k$ and I started to get terms that did look like $X(\nabla \times Y^i)$ which seemed promising, but I also got a lot of horrible junk terms looking like $\varepsilon^i_{jk}\frac{\partial X^a}{\partial x^j}\frac{\partial}{\partial x^a}Y^k$ which I don't know how to nicely take out of index notation, since just taking the partial derivative of the components of the vector field $X$ and feeding it $Y$ doesn't seem like a very geometric quantity to have.
Is there a better way of doing this, or is there some known identity someone has already found in a book that I can find?