Real and complex “boosts” in arbitrary pseudo-Euclidean spaces

For a special choice of the parameter matrices, pseudo-orthogonal and pseudounitary transformations are represented in the form of polynomials of second degree in these matrices. The connection between the transformation parameters and generalized velocities in the case of subluminal and superluminal generalized velocities is established. The resulting transformations are a generalization of Lorentz transformations describing pure motion to arbitrary real and complex pseudo-Euclidean spaces. The law of composition of the generalized velocities is found.