Consider the group $\operatorname{SL}_2(\mathbb{R})$. The set of positive-definite symmetric $2 \times 2$ matrices of determinant $1$ constitutes a symmetric space $S^+$ for $\operatorname{SL}_2(\mathbb{R})$ and may be identified with the quotient $\operatorname{SL}_2(\mathbb{R})/\operatorname{SO}_2(\mathbb{R})$. The Iwasawa decomposition implies that every matrix $g \in \operatorname{SL}_2(\mathbb{R})$ may be written as a product $g = uk$, where $u$ is a lower-triangular matrix and $k \in \operatorname{SO}_2(\mathbb{R})$. Thus, the set of lower-triangular matrices in $\operatorname{SL}_2(\mathbb{R})$ may be identified with $S^+$, via the map $u \mapsto uu^T$.
Question: Just as positive-definite symmetric $2\times2$ matrices of determinant $1$ may be identified with lower-triangular matrices in in $\operatorname{SL}_2(\mathbb{R})$, is there a similar sort of identification for indefinite symmetric $2 \times 2$ matrices of determinant $1$?
What I know: Let $S^0$ denote the set of indefinite symmetric $2 \times 2$ matrices of determinant $1$. Then $S^0$ also constitutes a symmetric space for $\operatorname{SL}_2(\mathbb{R})$ and may be identified with the quotient $\operatorname{SL}_2(\mathbb{R})/\operatorname{SO}(1,1)(\mathbb{R})$. I guess my question boils down to the following: is there an analogue of the Iwasawa decomposition that expresses every $g \in \operatorname{SL}_2(\mathbb{R})$ as $g = vh$, where $v$ belongs to some natural set of matrices in $\operatorname{SL}_2(\mathbb{R})$ and where $h \in \operatorname{SO}(1,1)(\mathbb{R})$?