Suppose I have objects which are rotations of d-dimensional real vectors with an additional optional scaling. Concatenating means multiplication of these objects. I want to define an addition operator, so that the "sum" of two rotations gives another unique rotation with scalings only. Which other assumptions do I need to show that only certain dimensions for the vectors are possible (under some conditions for division algebras d=2,4?; i.e. complex numbers and quaternions)? I suppose rotations already have some of the required properties?! Will the addition operation be neccessarily the one of complex numbers and quaternions?