suppose v = (v1,v2,v3,v4) in R^4 is perpendicular to (1,2,3,4) and (5,6,7,8).

then v.(a(1,2,3,4) + b(5,6,7,8)) = a(v.(1,2,3,4)) + b(v.(5,6,7,8)) = a0 + b0 = 0 + 0 = 0.

so it suffices to pick v perpendicular to each element of the basis {(1,2,3,4), (5,6,7,8)} of W.

you should be able to create a system of 2 linear equations in 4 variables, which will reduce to a system in two parameters.

chose those two parameters such that you get 2 linearly independent vectors in R^4, and you're done.