How would I prove the following theorem on quaternions?

Question

The theorem states

The map that takes $q$ to the map $[q] : x \to q^{-1}xq$ is a 2-to-1 homomorphism from the group of unit quaternions to $SO(3)$.

How would I prove this?

Answer 1

You

1. prove that the given map satisfies the axioms of a group homomorphism; then
2. prove that the kernel is $\{-1,1\}$.
   1. To accomplish that, you can note that if $[q]$ fixes, $i, j, k$, then it is in the center of $\mathbb H$. What is the intersection of the center of $\mathbb H$ with the unit quaternions?