How to check if a representation of SU(2) is irreducible I have found a representation $\rho$ of the group $G=\mathrm{SU}(2)$.
I want to show that this representation is irreducible but I don't know how.
Finding all invariant subspaces seems very difficult.
I verified that the condition:
$\forall g \in G \quad \rho(g) A=A \rho(g)$
implies that $A$ is proportional to the identity, but I couldn't find a result stating that this is enough for a representation to be irreducible.
Furthermore I know the character $\chi(g)$, but i do not know how this will help me, because the group under consideration is infinite.
Any pointers in the right direction would be greatly appreciated.

The condition you verified is indeed enough, this is due to complete reducibility for compact Lie groups. Roughly, suppose you have a subrepresentation then by complete reducibility it must have a complementary subrepresentation.  Now you can define a linear map that is the identity on one component and zero on the other, this will be a non-scalar map that commutes with the action of $G$.
I feel like I should put a disclaimer though: for Lie group  that is not compact this condition is not always enough, if complete reducibility fails you may have indecomposable but not irreducible objects such that the only intertwining endomorphisms are scalars.
