Suppose I have a representation $V$ of a finite group $G$, and decompose $V$ into $V = V_1 \oplus V_2$ where $V_1$ and $V_2$ are irreducible representations.
If I understand correctly, it is known that for any other decomposition $V = W_1 \oplus W_2$ into irreducibles $W_1, W_2$, we must have $V_1 \cong W_1$ and $V_2 \cong W_2$ (after reordering if necessary). However, I am having trouble seeing why this must be true.
I believe the idea is to apply Schur's Lemma to the identity map $I \colon V \to V$. The image $I(V_1)$ must be isomorphic to $V_1$ by Schur's Lemma, but I don't see how we can be sure that this image can be identified with one of the $W_i$ as opposed to some new subrepresentation $U \subset W_1 \oplus W_2$.