= Solution
Use the <Levi-Civita symbol> and sum over repeated indices. The <contraction of two Levi-Civita symbols> and the <product rule> give
$$
\begin{aligned}
[\nabla\times(F\times G)]_i
&=\epsilon_{ijk}\partial_j(\epsilon_{k\ell m}F_\ell G_m)\\
&=\partial_j(F_iG_j-F_jG_i)\\
&=F_i\partial_jG_j-G_i\partial_jF_j+G_j\partial_jF_i-F_j\partial_jG_i.
\end{aligned}
$$
Reading the four terms as <divergence> and directional derivatives proves the <curl of a cross product> identity
$$
\boxed{\nabla\times(F\times G)=F(\nabla\cdot G)-G(\nabla\cdot F)+(G\cdot\nabla)F-(F\cdot\nabla)G.}
$$
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