Let $N\in SU(2)$, $D^{(\frac{1}{2})}$ denote the fundamental representation of $SU(2)$ (i.e $D^{(\frac{1}{2})}(N)=N$ ) and let $\psi_{\alpha}\in\mathbb{C}^2$. Then $\psi_{\alpha}$ transforms under this irrep by $\psi_{\alpha}\rightarrow \psi'_{\alpha}=N_{\alpha}^{~\beta}\psi_\beta$ (summation convention in use). In general the $D^{(\frac{n}{2})}$ irrep acts on the space of totally symmetric spinors $\psi_{\alpha_1\dots\alpha_n}=\psi_{(\alpha_1\dots\alpha_n)}.$ I am trying to prove the well known result that $$D^{(\frac{n}{2})}\otimes D^{(\frac{m}{2})}=\sum_{i=|m-n|}^{m+n}\oplus D^{(\frac{i}{2})}.$$
I can do it for the low $m$ and $n$ cases by explicitly decomposing the vectorspace into invariant subspaces according to the above decomposition (and showing that they are indeed invariant). But how might I go about proving the result in general? I have read that you can do it using homogeneous polynomials and characters, but I would prefer it if I could do it by finding a general decomposition into invariant subspaces. Please do not post the answer, I only want suggestions, thank you.