Apply the <contraction of two Levi-Civita symbols> to the <cross product>:
$$
[A\times(B\times C)]_i=\epsilon_{ijk}\epsilon_{klm}A_jB_lC_m
=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})A_jB_lC_m.
$$
Thus the <vector triple product identity> is
$$
\boxed{A\times(B\times C)=B(A\cdot C)-C(A\cdot B).}
$$

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 12 / ii / Solution

 Solution (source code)