 Cross products with a common vector

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 ${ϵ}_{ijk}{ϵ}_{kpq}={\delta }_{ip}{\delta }_{jq}-{\delta }_{iq}{\delta }_{jp}$. The component of the double cross product becomes $A_i{ϵ}_{q\ell m}{C}_q{A}_{\ell }{B}_m-C_i{ϵ}_{p\ell m}{A}_p{A}_{\ell }{B}_m$. 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.