An amalgam uniqueness result for recognising q6:SU3(q)q6:SU3(q), G2(q)G2(q), or 3 ˙M10 using biaffine polar spaces

A biaffine polar space is formed by removing two hyperplanes, one from the polar space and one from the dual polar space. We describe two similar examples of these over a general field, each with the automorphism group acting flag-transitively. The stabilisers of certain flags are isomorphic in each geometry, hence they have similar amalgams. We classify such amalgams over finite fields, showing that only these two cases can occur when q≠2q≠2 and that they differ by whether certain subgroups commute. This leads to a recognition theorem for q6:SU3(q)q6:SU3(q), or G2(q)G2(q), for q≠2q≠2, with a third possibility 3 ˙M10 when q=2q=2.