A determinant on birational maps of Severi-Brauer surfaces

We define a determinant on the automorphisms of non-trivial Severi-Brauer surfaces. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this determinant and a group homomorphism found in [BSY23] we can determine the abelianisation of the Cremona group of a non-trivial Severi-Brauer surface. This is the first example of an abelianization of the Cremona group of a geometrically rational surface where the automorphism group is not trivial.