Conjugacy results for the Lie algebra sl₂ over an algebra which is a UFD

Let F be a field of characteristic not 2 and assume all algebras are over F. We establish several conjugacy theorems for the special linear Lie algebra sl_2 over an F-algebra which is a UFD. We find the structure of the full automorphism group as well as conditions for when two such algebras are isomorphic. We also study the structure of their derivation algebras. Often we work with more general coordinates when this if feasible.