I'm in the middle of trying to prove something at the moment and am looking for a decomposition of the Lie algebra $\mathfrak{su}(3)$ as a tensor product of $\mathfrak{su}(2)$ and something else, ie, an algebra $A$ such that $$ \mathfrak{su}(3) \simeq A \otimes \mathfrak{su}(2), $$ or some such result. Does anyone know of anything?