The study of magnons is an essential part of combining quantum information science and spintronics, allowing for the investigation of quantum properties such as entanglement in solid-state devices. Magnons are commonly described using the theory of T. Holstein and H. Primakoff, associating the spin operators with bosonic creation and annihilation operators. The quantum mechanical properties inherent to this description are the commutation relations. These relations must be conserved under transformation of the basis. This requires the application of pseudo-unitary transformations U (p, q) when studying the magnon eigenspectrum for example. Depending on the system at hand, the groups U (1, 1) and U (2, 2) are of particular interest and will be the focus of this work. We present a general formalism that leads to a representation of pseudo-unitary matrices via their self-adjoint elements. We apply this representation in examples involving magnons in antiferromagnets to find an explicit picture in connection to material properties. Finally, we explore numerical methods for determining magnon energies and compare them to the analytical counterpart.