We generalize an old result due to Lowenthal [1] and a more recent one due to Hamada [2] on the order of finite generation of the rotation group SO ( 3 ) both for fixed and arbitrary compound transformations. Unlike the above cited authors, we consider decompositions into factors with more than two invariant axes and provide rather intuitive geometric proofs. Thus, we derive a simple estimate for the number of factors in a decomposition and discuss possible means of optimization as well as particular examples of potential interest for the applications.