Prove Lagrange's Identity

Question

I am in need of major assistance with a homework problem I have been working on. It is to prove Lagrange's Identity, but by manipulating different forms of vector multiplication. The problem reads: (${A}\times{B}$) $\bullet$ (${C}\times{D}$) = (${A}\bullet{C}$)(${B}\bullet{D}$) - (${A}\bullet{D}$)(${B}\bullet{C}$). I have manipulated the dot and cross products into their scalar forms yielding the angle $\theta$ with either sine or cosine (respective to the dot or cross product). This has just lead me to a dead end and I do not know what I am even supposed to prove. Any help would be welcomed. I did find one post on another website, but they had suggested brute-forcing it by individual components. This seems like a ridiculous way to prove this, but I am open to solutions of all kinds.

Answer 1

The scalar quadruple product identity can be derived algebraically from the properties of tripe products.

$(A\times B)\cdot(C\times D) = D\cdot ((A\times B)\times C),~~~~\text{using shift property of scalar triple products}\\ =D\cdot (B(A\cdot C)-A(B\cdot C)),~~~~~~~~~~~~~~~~~~~~~~~\text{expansion of vector triple product}\\ =(D\cdot B)(A\cdot C)-(D\cdot A)(B\cdot C),~~~~~~~~~~~~~\text{linearity of dot product}\\ =(A\cdot C)(B\cdot D)-(A\cdot D)(B\cdot C).$