= Cross products with a common vector
{title2=$(A\times B)\times(A\times C)=(A\cdot(B\times C))A$}
The <cross product> of two <cross products> sharing their first vector is parallel to that vector, with scalar factor equal to the <scalar triple product>. In <suffix notation>, use $\epsilon_{ijk}\epsilon_{kpq}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}$. The component of the double <cross product> becomes $A_i\epsilon_{q\ell m}C_qA_\ell B_m-C_i\epsilon_{p\ell m}A_pA_\ell B_m$. The second term vanishes by antisymmetry of the <Levi-Civita symbol>; the first equals $A_i(A\cdot(B\times C))$, proving the identity even when the vectors are dependent.
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