Expanding a basis of a subspace to a basis for the vector space

You might take a different 2-D subspace $X = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_4, x_2 = -x_3\}$ which has a trivial intersection with $W$ and find a basis for it, for example $\{(0,2,-2,0),(1,0,0,1)\}$.

Hint: Any $2$ additional vectors will do, as long as the resulting $4$ vectors form a linearly independent set. Many choices! I would go for a couple of very simple vectors, check for linear independence. Or check that you can express the standard basis vectors as linear combinations of your $4$ vectors.