Construction of long root SL(2,q)-subgroups in black box groups

We present a one sided Monte--Carlo algorithm which constructs a long root $\sl_2(q)$-subgroup in $X/O_p(X)$, where $X$ is a black-box group and $X/O_p(X)$ is a finite simple group of Lie type defined over a field of odd order $q=p^k > 3$ for some $k\geqslant 1$. Our algorithm is based on the analysis of the structure of centralizers of involutions and can be viewed as a computational version of Aschbacher's Classical Involution Theorem. We also present an algorithm which determines whether the $p$-core (or "unipotent radical") $O_p(X)$ of a black-box group $X$ is trivial or not, where $X/O_p(X)$ is a finite simple classical group of odd characteristic $p$. This answers a well-known question of Babai and Shalev.