= Solution
Apply the <product rule> in <Einstein notation>, using the <Levi-Civita symbol> to express the <cross product>:
$$
\begin{aligned}
\nabla\cdot(\mathbf a\times\mathbf b)
&=\partial_i(\epsilon_{ijk}a_jb_k)\\
&=\epsilon_{ijk}(\partial_i a_j)b_k+\epsilon_{ijk}a_j\partial_i b_k\\
&=b_k\epsilon_{kij}\partial_i a_j-a_j\epsilon_{jik}\partial_i b_k\\
&=\mathbf b\cdot(\nabla\times\mathbf a)-\mathbf a\cdot(\nabla\times\mathbf b).
\end{aligned}
$$
The third line uses a cyclic permutation of indices in the first term and an interchange of two indices in the second. Rearranging proves the <divergence of a cross product> identity:
$$
\boxed{\mathbf a\cdot(\nabla\times\mathbf b)
=\mathbf b\cdot(\nabla\times\mathbf a)-\nabla\cdot(\mathbf a\times\mathbf b)}.
$$
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