  If a black box group is known to be isomorphic to an exceptional simple group
of Lie type of (twisted) rank $>1$, other than any $^2F_4(q)$, over a field of
known size, a Las Vegas algorithm is given to produce a constructive
isomorphism. In view of its timing, this algorithm yields an upgrade of all
known nearly linear time Monte Carlo permutation group algorithms to Las Vegas
algorithms when the input group has no composition factor isomorphic to any
group $^2F_4(q)$ or $^2G_2(q)$.
