I'm a bit confused about this following issue concerning representations of $SU(2)$.
Denote by 1 the 1-dimensional representation of the group $SU(2)$ (=the spin 0). Similarly, denote by 2 and 3 the 2-dimensional (spin 1/2) and 3-dimensional (spin 1) representations, respectively. Also, denote by $(j,j')$ the representation of $SU(2)\times SU(2)$ given by the tensor product of the $j$ representation with the $j'$ representation. By addition of angular momentum, we know that (2,2) = 1+3, where the + sign denotes the direct product of two representations. But since 1 and 3 both represent the same group $SU(2)$, so does their direct sum (a reducible representation). It follows that 1+3 is a representation of both $SU(2)$ and $SU(2)\times SU(2)$. Am I correct?