This is an exercise encountered in the book Algebra by M.Artin. Let $\mathbb{R}^3$ be the three-dimensional Euclidean space with coordinates $(x_1, x_2, x_3). $ Denote $H_i$ the subgroup of $SO_3$ that consists of rotations about the $x_i-$axis. In the first part of the problem, it is shown that $\forall P\in SO_3, P$ can be written as a product $P=A_1BA_2$, where $A_1, A_2$ are in $H_1$ and $B$ in $H_2. $ This representation is unique unless $B=diag\{-1,1,-1\} $or $I$, where $diag\{-1,1,-1\}=$ $diag\{1,-1,-1\}diag\{-1,1,-1\}diag\{1,-1,-1\}.$
The second part of this problem asks one to describe the double coset $H_1QH_1$ geometrically, where $Q$ is an element in $H_2$. Specifically, which rotation is in $H_1QH_1$ once Q is given. I think calculation using parametrized matrices is not clear enough, since we need to solve an eigenvector of eigenvalue 1 to determine the rotation axis of an element in $SO_3.$ Can anyone provide me with some ideas on this?