Vector Triple Production Reduction

[Illuminati]

$\nabla$x (A x B)
I'm trying to reduce this triple product down so nabla is only operating on either A or B, not the product of the two vectors.

I've been playing with this for a while now but can't really seem to make a dent in it. Any suggestions would be well appreciated.

 

[Tomer]

There's a pretty famous relation:

$\nabla\times(\nabla\times\vec{F}) = \nabla(\nabla \cdot F) - \nabla^{2}F$ =
= grad(div(F)) - Laplacian(F)

Proof can be of course rigorously done either through actual component calculation or using other relations, but I believe you can just use this :-)