= Triangular-rotation factorization of real determinant-one matrices
{title2=$g=hk,\quad h\text{ upper triangular},\quad k\in SO(2)$}
Every $g\in SL_2(\mathbb R)$ 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 $g=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ and $q=\sqrt{c^2+d^2}$, the positive branch is $h=\begin{pmatrix}1/q&(ac+bd)/q\\0&q\end{pmatrix}$ and $k=q^{-1}\begin{pmatrix}d&-c\\c&d\end{pmatrix}$.
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