Prove that (a x b) x (c x d) = [a, b, c]c - [a, b, c]d

Question

Prove that  \((\vec{a} \times \vec{b})\times (\vec{c} \times \vec{d}) = [\vec{a}, \vec{b}, \vec{c}]\vec{c} - [\vec{a}, \vec{b}, \vec{c}]\vec{d}\)

Answer 1

LHS = \((\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})\)