In this paper, we construct an explicit pairing in Cartier series for formal Lorentz groups of the form (X + Y + XY)/(1 + c²XY), where c is a unit of the ring of integers of the local field. We prove the basic properties of this pairing, namely, bilinearity and invariance, which make it possible to explicitly construct the generalized Hilbert symbol for formal Lorentz groups over rings of integers of local fields with the use of the obtained pairing.