Plane containing two points $a,b \in \mathbb{S}^3$ but avoiding two other antipodal points

Question

Let $\mathbb{S}^3$ be the three-dimensional subset of $\mathbb{R}^4$ given by $$ \mathbb{S}^3 = \{ (x_1,x_2,x_3,x_4) | x_1^2 + x_2^2 +x_3^2 +x_4^2 = 1 \}.$$

I want to construct a plane through the points $a,b\in \mathbb{S}^3$ while avoiding the points $x,y \in \mathbb{S}^3$ which are known to be antipodal. Is this always possible?