We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra ”dilatonic” coordinate, whose rotation, Lorentz and conformal groups are SO(d−1), SO(d−1, 1)×SO(1, 1) and SO(d, 2)×SO(2, 1), respective...