A Lie algebra method for rational parametrization of Severi–Brauer surfaces

It is well known that a Severi–Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi–Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.