This paper is a continuation of our previous paper viXra:2512.0150, in which we expressed all vectors of the 3D Euclidean vector space with operations involving rotations and homotheties of the unit vector of the x-axis. We defined also a new vector multiplication and in this paper we will show that with the usual vector addition, we have a K-algebra. We will also study the 4D case and we will also express all vectors with simple rotations and homotheties of the unit vector of the x-axis about 3 planes, we will define a vector multiplication analogous to what we have seen in the 3D case, and we will show that we have again a K-algebra with the usual vector addition. We will construct also a 4D fractal set which contains the 3D fractal set (which contains the Mandelbrot set) seen in our previous paper and we will show several 4D projections of that set in the 3D space. We will show also that there are no quaternion numbers, because there are actually geometrical operations involving the unit vector of the x-axis. In that case we will use compositions of 2 rotations of the unit vector of the x-axis about different planes to express all the vectors. In that case we will have also a division algebra. Finally, we will show that for all dimensions, we have K-algebras if we express all vectors with operations involving homotheties and simple rotations of the unit vector of the first axis (x_1), with the usual vector addition and a generalization of the multiplication that we have seen in our previous papers for the 2D and 3D cases and for the 4D case in this paper. One advantage of the simple rotations, among other ones, is that they allow to construct interesting n-dimensional fractal sets linked to the Mandelbrot set.