Direct sums and products of SU(2) representations

Question

I am reading the book on group theory and stuck with a simple problem. Why $$(2\bigotimes2)\bigoplus(2\bigotimes1)\bigoplus(1\bigotimes2)\bigoplus(1\bigotimes1)=3\bigoplus1\bigoplus2\bigoplus2\bigoplus1$$ is true? Here the digits denote irreducible representations of $SU(2)$ group of given dimension. As far as I know, $m\bigotimes n=(m+n)\bigoplus(m+n-1)\ ... \bigoplus|m-n+1|\bigoplus|m-n|$. So it follows that $2\bigotimes2=4\bigoplus3\bigoplus2\bigoplus1\bigoplus0$ and etc. So my answer is: $$4\bigoplus3\bigoplus2\bigoplus1\bigoplus0\bigoplus3\bigoplus2\bigoplus1\bigoplus3\bigoplus2\bigoplus1\bigoplus2\bigoplus1\bigoplus0$$ Could someone please point out the mistake? Thanks in advance.

Answer 1

We have $2 \bigotimes 2 = 3 \bigoplus 1$, $2 \bigotimes 1 = 1 \bigotimes 2 = 2$ and $1 \bigotimes 1 = 1$. Note how the identities would be valid if we hade ordinary times and plus (e.g. $2 \times 2 = 3 + 1$). This gives $$ (2 \bigotimes 2) \bigoplus (2 \bigotimes 1) \bigoplus (1 \bigotimes 2) \bigoplus (1 \bigotimes 1) = (3\bigoplus1) \bigoplus 2 \bigoplus 2 \bigoplus 1. $$