= Solution
In <Cartesian coordinates>, write $\partial_i=\partial/\partial x_i$ and use the <Levi-Civita symbol>. The <contraction of two Levi-Civita symbols> gives
$$
\begin{aligned}
[\nabla\times(\nabla\times\mathbf A)]_i
&=\epsilon_{ijk}\epsilon_{klm}\partial_j\partial_l A_m\\
&=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\partial_j\partial_l A_m\\
&=\partial_i\partial_jA_j-\partial_j\partial_jA_i.
\end{aligned}
$$
Here the <Kronecker delta> selects the indicated indices, and continuous second <partial derivatives> allow their order to be interchanged. Identifying the <gradient>, <divergence> and componentwise <Laplacian> proves the <curl of the curl identity>:
$$
\boxed{\nabla\times(\nabla\times\mathbf A)=\nabla(\nabla\cdot\mathbf A)-\nabla^2\mathbf A.}
$$
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