= Solution
In <suffix notation>, repeated indices are summed from one to three. The <Levi-Civita symbol> and <Kronecker delta> contraction is
$$
\epsilon_{ijk}\epsilon_{k\ell m}=\delta_{i\ell}\delta_{jm}-\delta_{im}\delta_{j\ell}.
$$
For distinct free index pairs this follows from the two possible matching permutations and their signs; if either pair repeats an index both sides vanish. Therefore
$$
\begin{aligned}
[a\times(b\times c)]_i
&=\epsilon_{ijk}a_j\epsilon_{k\ell m}b_\ell c_m\\
&=(\delta_{i\ell}\delta_{jm}-\delta_{im}\delta_{j\ell})a_jb_\ell c_m\\
&=b_i a_jc_j-c_i a_jb_j.
\end{aligned}
$$
Recognizing the <dot products> proves the <vector triple product identity>:
$$
\boxed{a\times(b\times c)=(a\cdot c)b-(a\cdot b)c.}
$$
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