Topological structure of $SO(4, \mathbb{R})$:

Regard $L \in SO(4, \mathbb{R})$ as a linear transformation on the space $\mathbb{R}^4$ of all quaternions. In particular, $L(1)$ is a non-zero quaternion and we may define the linear map $L^{\prime}: \mathbb{R}^4 \longrightarrow \mathbb{R}^4$ via the prescription

$$L^{\prime}(x) = L(x)L(1)^{-1},\quad x \in \mathbb{R}^4.$$