Cross products with a common vector ($(A\times B)\times (A\times C)=(A\cdot (B\times C))A$)

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 ${ϵ}_{ijk}{ϵ}_{kpq}={\delta }_{ip}{\delta }_{jq}-{\delta }_{iq}{\delta }_{jp}$. 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.