What is exactly the fourth dimension in Space?

The Fourth dimension in Space is a real distance dimension r, completing the three vector displacement dimensions. Space is a quaternion with one real dimension and three vector dimensions, s= r + Ix + Jy + Kz, where I,J and K denote the vector dimensions. The real dimension r is related to time by r=ct where c is the speed of light.

Einstein and Minkowski proposed in Relativity Theory a defective four dimension Space, where the fourth dimension is a vector Ict . This Space is essentially a two dimensional Space and does not exhibit the non-commuttive features of the quaternion Space.