In Für Elise by Beethoven, in many variations found on the net, starting from bar 13, and as I have seen in many videos playing this part, both hands are used consecutively while most to all of them can be easily played with just the right hand. Is there any special reason why this happens (something like making it look pretty)? Is it written by Beethoven himself or have later editors made these changes? Is it wrong or considered rude (because great pianists have not done so) if I play it with only one hand?
PE: My problem is not where E has different octaves, but those D# & Es and continuous moving from the bottom staff to the top staff starting from bar 13, is my main problem, and yes about those Es: although that E to E travel could be done using both hands differently, for example it's much easier for me to play the starting E in bar 13 with my left pinky, the second E with my left thumb and the rest with my right hand, and the highest E with my right hand.