I came across the definition of a completely reducible representation of a group. The way I understand the definition there is no real difference between a reducible representation and a completely reducible representation. The book I'm currently reading states that "This means that a completely reducible representation can be written, with a suitable choice of basis, as the direct sum of irreducible representations.". But, as far as I know, any representation is either reducible or irreducible so, eventually, any reducible representation can be written as a direct sum of irreducible representations.
What am I missing? What is the difference between reducible and completely reducible?