There is a way to show the invarience of space-time in a very straight forward manner:
Given the Lorentz transformations:
dt = Ƴ( dt' + (v*dx')/(c^2) )
dx = Ƴ( dx' + v*dt' )
Ƴ = 1/(1-(v/c)^2)^.5
-> dt and dx denote finite changes in the S frame, dt' and dx' denote changes in the S'...