= Solution
Use the <Levi-Civita symbol> and summation over repeated indices. Contracting the two symbols gives
$$
\begin{aligned}
[\nabla\times(\mathbf F\times\mathbf G)]_i
&=\epsilon_{ijk}\partial_j(\epsilon_{klm}F_lG_m)\\
&=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\partial_j(F_lG_m)\\
&=\partial_j(F_iG_j-F_jG_i)\\
&=F_i\partial_jG_j-G_i\partial_jF_j+G_j\partial_jF_i-F_j\partial_jG_i.
\end{aligned}
$$
Thus the <curl of a cross product> is
$$
\boxed{\nabla\times(\mathbf F\times\mathbf G)
=\mathbf F\nabla\cdot\mathbf G-\mathbf G\nabla\cdot\mathbf F
+(\mathbf G\cdot\nabla)\mathbf F-(\mathbf F\cdot\nabla)\mathbf G.}
$$
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