= Solution
For differentiable <vector fields>, expand the <divergence> using the <Levi-Civita symbol> and the product rule:
$$
\begin{aligned}
\nabla\cdot(F\times G)
&=\partial_i(\epsilon_{ijk}F_jG_k)\\
&=\epsilon_{ijk}(\partial_iF_j)G_k+\epsilon_{ijk}F_j\partial_iG_k\\
&=G\cdot(\nabla\times F)-F\cdot(\nabla\times G).
\end{aligned}
$$
The first contraction uses the cyclic identity $\epsilon_{ijk}=\epsilon_{kij}$; the second uses $\epsilon_{ijk}=-\epsilon_{jik}$. This fixes the relative minus sign without guessing the orientation of the <cross products>.
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