  Given a black box group $\mathsf{Y}$ encrypting $\rm{PSL}_2(\mathbb{F})$ over
an unknown field $\mathbb{F}$ of unknown odd characteristic $p$ and a global
exponent $E$ for $\mathsf{Y}$ (that is, an integer $E$ such that
$\mathsf{y}^E=1$ for all $\mathsf{y} \in \mathsf{Y}$), we present a Las Vegas
algorithm which constructs a unipotent element in $\mathsf{Y}$. The running
time of our algorithm is polynomial in $\log E$. This answers the question
posed by Babai and Beals in 1999. We also find the characteristic of the
underlying field in time polynomial in $\log E$ and linear in $p$.
  Furthermore, we construct, in probabilistic time polynomial in $\log E$,
  1. a black box group $\mathsf{X}$ encrypting $\rm{PGL}_2(\mathbb{F})
\cong\rm{SO}_3(\mathbb{F})$, its subgroup $\mathsf{Y}^\circ$ of index $2$
isomorphic to $\mathsf{Y}$ and a probabilistic polynomial in $\log E$ time
isomorphism $\mathsf{Y}^\circ \longrightarrow \mathsf{Y}$;
  2. a black box field $\mathsf{K}$, and
  3. polynomial time, in $\log E$, isomorphisms \[ \rm{SO}_3(\mathsf{K})
\longrightarrow \mathsf{X} \longrightarrow \rm{SO}_3(\mathsf{K}). \]
  If, in addition, we know $p$ and the standard explicitly given finite field
$\mathbb{F}$ isomorphic to $\mathbb{F}$ then we construct, in time polynomial
in $\log E$, isomorphism \[ \rm{SO}_3(\mathbb{F})\longrightarrow
\rm{SO}_3(\mathsf{K}). \]
  Unlike many papers on black box groups, our algorithms make no use of
additional oracles other than the black box group operations. Moreover, our
result acts as an $\rm{SL}_2$-oracle in the black box group theory.
  We implemented our algorithms in GAP and tested them for groups such as
$\rm{PSL}_2(\mathbb{F})$ for $|\mathbb{F}|=115756986668303657898962467957$ (a
prime number).
