A minimal requirement for different inertial observers to be equivalent is that each one is perceived by the other as existing always in the past and in the future. This aspect is formalized in the notion of mutual objective existence. Using just this notion, we show that, in the bidimensional case (x 0, x 1), the linear transformations A(x 0, x 1) connecting two different frames form a Lorentz group (or its contractions, Galilei and Carroll). In three dimensions (one time) x 0 and two space variables x 1 and x 2) the transformations compatible with the mutual objective existence are the product of A 1(x 0, x 1)A 2(x 0, x 1), where both A 1 and A 2 are one of the previous transformations, times a space transformation R(x 1,x 2), which is obliged to be Euclidean when both A 1 and A 2 are Lorentzian.