Tensor product of simple $sl_2$ modules

I am working on the following problem:
Let $M(n)$ be the finite-dimensional, simple $\mathfrak{sl}_2(\mathbb{C})$-module with highest weight $n\in\mathbb{N}_0$. Show that the module $M(l)\otimes_{\mathbb{C}} M(k)$ is cyclic for $l,k\in\mathbb{N}_0$.
I am familiar with the Clebsch-Gordan statement. Hence, I am puzzled by how the dimension of the tensor product can be 1:$$M(l)\otimes M(k)\cong M(l+k)\oplus M(l+k-2) \oplus … M(|l-k|)$$for $k\geq l$. What am I overlooking? Any help is greatly appreciated!