Question on vector cross product. Show that $\big((\mathbf{u} \times (\mathbf{u}\times \mathbf{v})) \times \mathbf{v}\big) \times (\mathbf{u} \times \mathbf{v})=0$.

From wolfram, it gives zero but there's no details. How to prove this?
Using the identity
$$\textbf{A}\times (\textbf{B}\times \textbf{C})= (\textbf{A}.\textbf{C})\textbf{B}-(\textbf{A}.\textbf{B})\textbf{C}$$
we get,
$\begin{align}
((\textbf{u}\times(\textbf{u}\times \textbf{v}))\times \textbf{v})\times(\textbf{u}\times \textbf{v})\\&amp;=(((\textbf{u}.\textbf{v})\textbf{u}-(\textbf{u}.\textbf{u})\textbf{v})\times\textbf{v})\times(\textbf{u}\times \textbf{v})\\&amp;=((\textbf{u}.\textbf{v})(\textbf{u}\times\textbf{v})-(\textbf{u}.\textbf{u})(\textbf{v}\times\textbf{v}))\times(\textbf{u}\times \textbf{v})\\&amp;=(\textbf{u}.\textbf{v})(\textbf{u}\times\textbf{v})\times(\textbf{u}\times \textbf{v})=0\\
\end{align}$
Hint:
$a\times b=0$ if and only if $a$ and $b$ are parallel.