Triangular-rotation factorization of real determinant-one matrices ($g=hk,h\text{ upper triangular},k\in SO(2)$)

Every $g\in S{L}_2(\mathbb{R})$ factors into an upper triangular determinant-one matrix and a rotation matrix. This follows because the triangular subgroup acts transitively on the complex upper half-plane, a group orbit of the full group, while $SO(2)$ is the stabilizer subgroup of $i$. The intersection of the two subgroups is $\{\pm I\}$, giving exactly two matrix pairs $(h,k)$ and $(-h,-k)$. Requiring the diagonal of $h$ to be positive gives uniqueness. For $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and $q=\sqrt{c^2+d^2}$, the positive branch is $h=\left(\begin{array}{cc}1/q& (ac+bd)/q\\ 0& q\end{array}\right)$ and $k=q^{-1}\left(\begin{array}{cc}d& -c\\ c& d\end{array}\right)$.