In July 1905 (published in January 1906) Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the principle of least action ; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group, and he showed that the combination x +y +z -t is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing c t − − 1 as a fourth imaginary coordinate, and he used an early form of four-vectors. He also formulated the velocity addition formula, which he had already derived in unpublished letters to Lorentz from May 1905: