Solution
= Solution
The <vector triple product identity> and $\mathbf n\cdot\mathbf n=1$ give
$$
H^2\mathbf x=\alpha^2\mathbf n\times(\mathbf n\times\mathbf x)=\alpha^2\bigl[\mathbf n(\mathbf n\cdot\mathbf x)-\mathbf x\bigr].
$$
Apply $H$ once more. The term parallel to $\mathbf n$ vanishes under its <cross product> with $\mathbf n$, leaving $H^3\mathbf x=-\alpha^3\mathbf n\times\mathbf x=-\alpha^2H\mathbf x$. The result includes $\alpha=0$.