SCHWEITZER 2012 Problem 5

Let $V_1,V_2,V_3,V_4$ be four dimensional linear subspaces in $\mathbb{R}^8$ such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace $W$ in $\mathbb{R}^8$ such that all four vector spaces $W\cap V_i$ are two dimensional.