There is a natural way to identify $\mathbb{R}P^3:=S^3/\mathbb{Z}_2$ to $SO(3):=\{ A\in GL(3)\ | A^tA=I_3,\ \det(A)=1\}$. To a point $[(w,x,y,z)]\in\mathbb{R}P^3$ associate the $3D$ rotation through the axis generated by $(x,y,z)\in\mathbb{R}^3$ of angle $2\theta$, where $\cos\theta=w$ (and $|\sin\theta|=\sqrt(x^2+y^2+z^2)$). One can see an explicit formula for this map on the Wikipedia page dedicated to 3D rotations https://en.wikipedia.org/wiki/3D_rotation_group in the paragraph "Connecting $SO(3)$ and $SU(2)$".
Moreover there is a natural action of $GL(4)$ on $\mathbb{R}P^3$, i.e. $W\cdot ([w,x,y,z)])=[W(w,x,y,z)/\||W(w,x,y,z)\||]$.
My question is the following : does someone has a "nice formula" of this action of $GL(4)$ on $SO(3)$ through the identifications used in the previous paragraph?
Actually I only need to understand the action of $Sp(4)$, i.e. the symplectic group (maybe even only the diagonal matrices of $Sp(4)$ on $SO(3)$).
Thanks in advance.
