In 1968, A. Roy defined a set of transformations which generalises the classical Eichler-Siegel-Dickson transformations over fields to commutative rings. These elementary orthogonal transformations are defined on quadratic spaces with a hyperbolic summand over a commutative ring with unity in which 2 is an invertible element. These transformations are called Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal transformations and the group generated by them is called the DSER elementary orthogonal group. In this talk I will compare the DSER elementary orthogonal group and the odd elementary hyperbolic unitary group introduced by V.A. Petrov in 2005. We prove that these two groups over a commutative ring with a pseudo-involution are conjugate subgroups of the general linear group.
