Proof:

We show that the map $\phi: SO(4, \mathbb{R}) \longrightarrow S^3\times O(3, \mathbb{R})$ given by $\phi(L) = (L(1), L^{\prime})$, where $L^{\prime}$ is defined as in the previous lemma, is a homeomorphism. The map $L^{\prime}$ is an element of $O(3, \mathbb{R})$ since it maps $\Pi$ to itself and preserves Euclidean norm. Further, $L(1)$ is obviously a unit quaternion. The image of $\phi$ is a compact connected subspace of $S^3\times O(3, \mathbb{R})$ and sends the identity element to the pair $(1,$   id$)$ which means the image must be contained in $S^3\times SO(3, \mathbb{R})$. It is an exercise that the map is bijective. Since the space $SO(4, \mathbb{R})$ is compact and $S^3\times SO(3, \mathbb{R})$ is Hausdorff, it follows that $\phi$ is a homeomorphism.