Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch--Gordan decomposition then gives that $$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$ But suppose I want to think of this decomposition as matrices. Evaluating at a point $x \in \mathrm{SU}(2)$, on the left I have $$ (\pi_m(x))_{ij} (\pi_n(x))_{pq}.$$ How do the indices $i,j$ and $p,q$ correspond to the indices on the big matrix on the right?
Some motivation for the question. Expressions involving products of Fourier coefficients arise naturally in nonlinear harmonic analysis on $\mathrm{SU}(2)$. In this specific instance I am interested in calculating $\operatorname{Tr}(\hat{f}(\pi_m) \pi_m(x)) \operatorname{Tr}(\hat{f}(\pi_n)\pi_n(x)) = \hat{f}(\pi_m)^{ji} \hat{f}(\pi_n)^{qp} \pi_m(x)_{ij} \pi_n(x)_{pq}$ using the Clebsch--Gordan expansion above. Here $\hat{f}(\pi) = \int_{\mathrm{SU}(2)} f(x) \pi(x^{-1}) \mathrm{d} x. $