The algebraic decomposition of a unitary operator is a key operation in the synthesis of quantum circuits. If most methods factorize the matrix into products, there exists a method that allows to reuse already existing optimized circuits to implement linear combinations of them. This paper presents an attempt to extend this method to a general framework of circuit synthesis. The method needs to find suitable groups for the implementation of new quantum circuits. We identify key points necessary for the construction of a comprehensive method and we test potential group candidates.