Consider the following sets $R$ and $S$ of quadruples of rational numbers:

$R = \left\{\left(\frac{1}{3},\frac{-2}{3},\frac{1}{3},0\right),\left(\frac{-3}{2},0,\frac{3}{2},0\right)\right\}$

$S = \{(1,0,-1,0),(1,-1,0,0),(-1,1,0,1),(1,0,-1,1)\}$.

If $V$ (respectively, $W$) is the subspace of the rational quadruples spanned by $R$ (respectively, $S$), then the subspace $U$ of the rational quadruples satisfying the following two conditions

$W = U \oplus V$ (i.e., $W$ is the direct sum of $U$ and $V$)

$U \perp V$ (i.e., $U$ is orthogonal to $V$ with respect to the standard inner product of $\mathbb{Q}$)

is ___.