If I expand the following expression for the 3 components I see that the expression vanishes, $|\mathbf{n}|=1$: $$ \left(\mathbf{n}\times (\mathbf{n}\times \mathbf{x})\right)\cdot \mathbf{x} +\left(\mathbf{n}\times (\mathbf{n}\times \mathbf{x})\right)\cdot \left(\mathbf{n}\times \left(\mathbf{n}\times \mathbf{x}\right)\right)=\mathbf{0} $$
Is there a more elegant proof?
EDIT A simple picture also give a proof. I'm looking for a algebraic proof, like some manipulation using vector-algebra.
Hint:
Let ${\bf a}\bot {\bf n}$ with $|{\bf a}|=1$ and ${\bf b}={\bf n}\times {\bf a}$. Then if the expression vanishes for ${\bf x}\in\{ {\bf a},{\bf b},{\bf n}\}$ then you are done since they constitue a basis of $\mathbb{R}^3$