Dimension of intertwining space of unitary representation I'm currently trying to read through an article by Poguntke, to be found here. The main theorem of the article is the following:

Theorem. Let $\pi$ and $\pi'$ be irreducible unitary representations of the group $G$ in the Hilbert spaces $H$ and $H'$, respectively, and let $\rho$ be a unitary representation of $G$ in a Hilbert space of dimension $n<\infty$. Then the dimension of the space $\mathrm{Hom}_G(\pi',\rho\otimes\pi)$ of intertwining operators is not greater than $n$.

With this theorem, the author proves the following:

Corollary 1. Let $G$, $\pi$, $\pi'$, $H$ be as in the theorem. Then the dimension of the algebra $\mathrm{Hom}_G(\rho\otimes\pi,\rho\otimes\pi)$ is not greater than $n^2$. Especially, $\rho\otimes\pi$ is the direct sum of at most $n^2$ irreducible subrepresentations.

The first part of the corollary is easy, but I don't get why the second part (in italics) holds. As I read it, there exist irreducible subrepresentations $\alpha_1,\ldots,\alpha_m$ of $\rho\otimes\pi$ such that $\rho\otimes\pi=\bigoplus_{i=1}^m\alpha_i$ and $m\leq n^2$, but how do these subrepresentations arise?
 A: $\def\Hom{\mathop{Hom}}$
So here's how you prove the corollary. I'll prove the more general statement that if $V$ is a representation such that 
$$\Hom_{G}(V,V) \le n > 0$$
then $V$ is a direct sum of at most $n$ irreducible subrepresentations.
If $V$ is irreducible, we are done. So suppose $V$ is not irreducible. Then, $V$ has some subrepresentation $V_{1}$ and hence, we have
$$V = V_{1} \oplus V_{2}$$
with $V_{2} = V_{1}^{\perp}.$
Again, if $V_{1}$ and $V_{2}$ are both irreducible, we are done. Otherwise, without loss of generality $V_{2}$ is not irreducible and hence, we have, a subrepresentation $V_{3}$ and a complement $V_{4} = V_{3}^{\perp} \cap V_{2}.$ This gives us
$$V = V_{1} \oplus V_{3} \oplus V_{4}.$$
Continue doing this until either
(a) All the subrepresentations in the summation become irreducible.
or
(b) There are $n+1$ or more subrepresentations.
If (a) happens, we are done. So suppose (b) happens. Then, we have
$$V = V_{1} \oplus \cdots \oplus V_{m}$$
with each $V_{i}$ a subrepresentation and $m > n.$ But then $\Hom_{G}(V, V)$ contains
$$\oplus_{i=1}^{m} \Hom_{G}(V_{i}, V_{i})$$
which has dimension greater than or equal to $m > n$, a contradiction. Hence (b) cannot happen, and thus the corollary is true.
As for how to actually find the subrepresentations, I am not sure.
