Natural representations of black box groups encrypting $S{L}_2({\mathbb{F}}_q)$

Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\mathrm{S}\mathrm{L}}_2(\mathbb{F})$, where $\mathbb{F}$ is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in $\log E$, the isomorphisms $$\mathsf{Y}⟷{\mathrm{S}\mathrm{L}}_2(\mathsf{K}),$$ where $\mathsf{K}$ is a black box field encrypting $\mathbb{F}$. Our algorithm makes no reference to any additional oracles. We also give similar algorithms for black box groups encrypting ${\rm PGL}_2($\mathbb{F}$)$,${\rm PSL}_2($\mathbb{F}$)$.