 In the following we shall use imaginary time (formula_1) so that space-time is Euclidean and the equations are simpler.The Poincaré transformations are given by the coordinate transformation on space-time parametrized by the 4-vectors "V"

where formula_3 is an orthogonal matrix and formula_4 is a 4-vector.Applying this transformation twice on a 4-vector gives a third transformation of the same form.