An algorithm for Lang's Theorem

We give an efficient Las Vegas type algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest. For our time analysis we derive that the proportion of reflection derangements in a Weyl group is less than 2/3.