Q:

Quadruple product

Looking to prove the following:
$\langle a\times b,c\times d\rangle
=\langle a,c\rangle \langle b,d\rangle
-\langle a,d\rangle \langle b,c\rangle$
Where $\langle ,\rangle$ and $\times$ denote the dot product and cross product resp. I'm wondering if there's a way to go about this without simply grinding out the algebra.
Any help is appreciated.

A:

First substitute for $$a \times b = e$$ then we get,
$$
<a\times b, c \times d> \quad = \quad < e, c \times d> \\
< e, c \times d> \quad \Rightarrow \quad e \cdot (c \times d) = d \cdot (e \times c) \\
\\
d \cdot (e \times c) = \quad d \cdot (a \times b \times c) \\
\qquad \qquad \qquad = d \cdot [ (a \cdot c)b - (b \cdot c)a ] \\
\qquad \qquad \qquad \quad \quad = (d \cdot b)(a \cdot c) - (a \cdot d)(b \cdot c) \\
\qquad \qquad \qquad \qquad \qquad \qquad  = <a,c><d,b> - <b,c><a, d>
$$
Take a look at .. Dot-Cross Identities

