AbstractWe exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank$2$in$\mathrm{PSL}_2(\mathbb{R})$or$\mathrm{SL}_2(\mathbb{R})$. This algorithm, together with methods for checking whether a two-generator subgroup of$\mathrm{PSL}_2(\mathbb{R})$or$\mathrm{SL}_2(\mathbb{R})$is discrete and free, have been implemented inMagma for groups defined over real algebraic number fields.Supplementary materials are available with this article.