Linear Transformations · Challenge problem

Let $\mathbf a = (1,2,2)$ and let $T:\mathbb{R}^3\to\mathbb{R}^3$ be $T(\mathbf v) = \mathbf v\times\mathbf a$. There is a number $k$ with $T^3 = kT$ (that is, $T(T(T(\mathbf v))) = k\,T(\mathbf v)$ for every $\mathbf v$). Find $k$.