Linear Transformations · Superchallenge problem

In $\mathbb{R}^4$ let $U = \operatorname{span}\{(1,1,0,0),\ (0,0,1,1)\}$ and $W = \operatorname{span}\{(1,1,0,0),\ (1,0,1,0)\}$. The linear maps $T:\mathbb{R}^4\to\mathbb{R}^4$ with $T(U)\subseteq U$ and $T(W)\subseteq W$ form a subspace of the $16$-dimensional space of all linear maps $\mathbb{R}^4\to\mathbb{R}^4$. What is its dimension?