Prove that $SO_{3}/D$ is a rational variety over $\mathbb{Q}$ where $D=Q_{8}/\{-1,+1\}$

Question

I am reading on my own from Topics in Galois Theory by Serre.

I am stuck at the below mentioned fact, I can directly use it assuming it's true, but I want to understand the proof. Since I am learning Algebraic Geometry also, It will make my concepts rigid.

I want to prove that that $SO_{3}/D$ is a rational variety over $\mathbb{Q}$ where $D=Q_{8}/\{-1,+1\}$.

I would love to read a detailed answer. Thanks in advance