(a x b) x (c x d) can be expanded two ways:

Method 1: Let p = a x b. Then p x (c x d) = (p.d)c - (p.c)d = [a b d]c - [a b c]d.

Method 2: Let q = c x d. Then (a x b) x q = (a.q)b - (b.q)a = [c d a]b - [c d b]a = [a c d]b - [b c d]a.

Both expressions are equal: [a b d]c - [a b c]d = [a c d]b - [b c d]a.

This gives the identity: [a b d]c - [a b c]d = [a c d]b - [b c d]a, which can be rearranged to express any one vector in terms of the other three (when [b c d] != 0).