There are several methods, which can double the size of an existing magic square. With small changes, basic magic squares of odd and even order can be processed by these methods.
|Benson - Jacoby|
|Strachey (Benson-Jacoby Variant)|
|Widdis - Richter|
All algorithms are described in detail in chapter Magic Squares of Doubled Order (german: Verdopplung der Ordnung) together with some other PDF documents.