= Solution
Write both <cross products> with the <Levi-Civita symbol>. Its contraction identity gives
$$
[a\times(b\times c)]_i=\epsilon_{ijk}\epsilon_{klm}a_jb_lc_m
=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_jb_lc_m
=b_i(a\cdot c)-c_i(a\cdot b).
$$
Therefore the <vector triple product> identity is
$$
\boxed{a\times(b\times c)=(a\cdot c)b-(a\cdot b)c.}
$$
The order matters: the <cross product> is not associative, and exchanging the two nested arrangements changes the expression.
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