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How to simplify $(u\times v)\cdot (v\times w)\times (w\times z)$ without brute force ($u,v,w,z\in \mathbb{R}^3$)? With only basic properties of the two products.

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Let $[a,b,c]$ be a shorthand for the triple product $a \cdot (b \times c)$. Using the identities $$a \times (b \times c) = (a\cdot c)b - (a\cdot b)c \quad\text{ and }\quad [a,b,c] = [b,c,a] = [c,a,b] $$ we have $$\require{cancel}(v \times w )\times(w \times z) = ((v \times w)\cdot z) w - \color{red}{\cancelto{\,0}{\color{gray}{((v \times w)\cdot w)}}}z = [v,w,z] w $$ This leads to $$\begin{align}[u\times v,v \times w,w \times z] &= (u\times v)\cdot ([v,w,z]w) = [v,w,z] (w\cdot (u \times v))\\ &= [v,w,z][w,u,v] = [u,v,w][v,w,z] \end{align} $$

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