$$\vec{B}\cdot \operatorname{curl} \vec{A}-\vec{A}\cdot \operatorname{curl} \vec{B} = \operatorname{div}(\vec{A}\times \vec{B})$$
So I started from the LHS as follows:
$$=B_i\epsilon_{ijk}\frac{\partial}{\partial x_j} A_k-A_i\epsilon_{ijk}\frac{\partial}{\partial x_j}B_k$$
$$= \epsilon_{ijk}\left(B_j\frac{\partial A_k}{\partial x_j} - A_i\frac{\partial B_k}{\partial x_j}\right)$$
I'm not sure if I've done these steps correctly, but assuming I have; I can't see where to go from here.
I know that $$\operatorname{div}\vec{A}\times\vec{B} = \frac{\partial}{\partial x_i}(\epsilon_{ijk}A_jB_k)$$
By the way I'm new to tensor notation, so please explain things you might think are obvious (because they probably aren't obvious to me). Thanks