## Levi Civita proof - Curl

I need to prove that $$\nabla \times (\vec{A} \times \vec{B})$$

Well. Trying to solve it, I've come to this:

$$\partial_j \vec{A}_i \vec{B}_j \hat{u}_i- \partial_i \vec{A}_i \vec{B}_j \hat{u}_j$$

Then I've found in my book that this is equal to:

$$\vec{A}_i \partial_j \vec{B}_j \hat{u}_i + \partial_j \vec{B}_j \vec{A}_i \hat{u}_i - (\vec{A}_i \partial_i \vec{B}_j \hat{u}_j + \vec{B}_j \vec{A}_i \partial_i \hat{u}_j)$$

Can someone explain to me why?