TITLE: What is the difference between a reducible representation and a completely reducible representation?
QUESTION [7 upvotes]: 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?
Thanks.

REPLY [4 votes]: Def.: A reducible representation $(\rho,V)$ has a(t least one) sub representation $(\rho_W,W)$, where W a subset of V and W is closed under the action of $\rho$.
Def.: Completely reducible means that there are subsets $W_i$ of V such that each $W_i$ forms together with its $\rho_{W_i}$ a sub representation and $V=\oplus_{i} W_i$.
As @Derek Holt mentioned in the comments a counterexample to your statement that every reducible representation can be written as a direct sum of irreducible is the following: $$x \mapsto \left(\begin{array}{cc}1&x\\0&1\end{array}\right)$$ You can check that $$a\left(\begin{array}{c}1\\0\end{array}\right)$$, where a is a real number is an irreducible sub representation but $\left(\begin{array}{cc}1&x\\0&1\end{array}\right)$ cannot be written as a direct sum of irreducible sub representations because it is not diagonalizable (det=1).