Semisimplicity of a group representation Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}\,$:
$$
 A:\;\;G\,\longrightarrow\,GL({\mathbb{V}})\;\;,
 $$
and let ${\mathbb{V}}$ be decomposable into a finite or infinite direct sum
$$
 {\mathbb{V}}\,=\,\bigoplus_{i\in\cal I} {\mathbb{V}}_i
 $$
of invariant subspaces carrying irreducible subrepresentations of $A\,$:
$$
 A(G)\,{\mathbb{V}}_i\subseteq{\mathbb{V}}_i\;\;,\qquad A(G)\,=\,\bigoplus_i \underbrace{A_i(G)}_{  \stackrel{\;}{\textstyle{\rm{irreducibles}}}   }
 \;.
 $$
On page 23 of his book "Representation Theory" (page 27 of the pdf file), Kowalsky issues a warning that the said decomposition does not imply that any subrepresentation is a subsum
$$
\bigoplus_{i\in\cal J} {\mathbb{V}}_i\;\;,\quad{\cal J}\subset{\cal I}\;.
$$
The author emphasises that this would be false even for $G$ trivial, where the only irreducible representation is the trivial one, and writing a decomposition would amounts to choosing a basis. He then states that there are usually many subspaces of $\mathbb V$ which are not literally direct sums of a subset of the basis directions.
Could someone please offer me a simple example or two, illustrating the author's point?
 A: For instance, imagine that $G$ has an irreducible action on a vector space $W$. Then $G$ acts naturally into $V=W \oplus W$ (and this the decomposition of $V$ into irreducible subrrepresentations). Here, the subspace $U = \{(w, w) \mid w \in W\}$ is clearly a $G$-invariant subspace, but it is not a sum of $W \oplus \{0\}$ and $\{0\} \oplus W$.
The concrete example the author is mentioning is with $G=\{e\}$, for which a $G$-action is trivial. Therefore, the irreducible spaces are all of the form $\mathbb{C}$, and fixing such a decomposition is the same as writting $V\simeq \mathbb{C}^n$. The problem is, $\mathbb{C}^n$ admits a lot of subspaces, all of them being subrepresentations. And of course not all of them are of the form $\bigoplus_{I \subseteq \{1, \dots, n\}} \mathbb{C}$.
Edit: For these examples we do need repeated irreducible representations. To illustrate this point, imagine that $V=W_1 \oplus W_2$ is a direct sum of two non-isomorphic subrepresentations, and suppose that $W \subseteq V$ is an irreducible subrepresentation, different from $0, W_1, W_2$ and $W$. Note that the projection $W \to W_1$ is either $0$ or an isomorphism, thanks to Schur's Lemma. If it is zero, then $W \subseteq W_2$, a contradiction, and so $W \cong W_1$. But using the other projection, we get that $W \simeq W_2$, which contradicts that $W_1$ and $W_2$ are not isomorphic.
