Identity for the cross of a curl?

[center o bass]

Hello! I'm want to prove a vector identity for

$$(\nabla \times \vec{A}) \times \vec A$$

using the familiar method of levi-civita symbols and the identity

$$\epsilon_{kij}\epsilon{kmn} = \delta_{im}\delta_{jn} - \delta_{in}\delta{jm}$$,
but I don't seem to come up with any usefull answer. I end up with that


$$[(\nabla \times \vec{A}) \times \vec A]_k = (\partial_j A_k)A_j - (\partial_k A_j)A_j$$
, which doesn't seem to reduce something familiar in terms of vectors and vector operators. Any idea how I might get there?

 

[fresh_42]

I don't think you can essentially simplify it:
https://en.wikipedia.org/wiki/Vector_calculus_identities
https://de.wikipedia.org/wiki/Rotation_eines_Vektorfeldes#Rechenregeln