Proof of invariance of dp1*dp2*dp3/E

[ala]

If we introduce four-dimensional coordinate sistem with component of four-momentum on axis, then dp1*dp2*dp3 can be considered as zeroth component of an element of hypersurface given by p^2=m^2*c^2. Element of this hypersurface is parallel to 4-vector of momentum so we have that dp1*dp2*dp3/E is ration of components of two parallel vectors so it must be invariant under Lorentz transformation. (this proof is from Landau, Lifgarbagez - Fields)
This proof seems ok but I can't see why I cannot apply it to show (wrongly) that dxdydz/t is invariant.

Thanks in advance.

 

[Meir Achuz]

The proof for momentum and energy uses the constraint E^2=p^2+m^2.

 

[ala]

Yes, that is equivalent to p^2=m^2*c^2 (p is 4-vector of momentum). So for x, I can watch hypersurface given by equation x^2=c*\tau (x is 4-vector, and \tau proper time). So the rest is same and conclusion is worng (namely that dxdydz/t is invariant)