Let $\mathfrak{g} = \mathfrak{sl}(2, \mathbb{C})$. Every irreducible representation of $\mathfrak{sl}(2, \mathbb{C})$ has a form of $V_{n} = \mathrm{Sym}^{n}(V_{1})$, where $V_{1} = \mathbb{C}^{2}$ is a standard representation. One can prove that the tensor product $V_{m}\otimes V_{n}$ is decomposed as $$ V_{m}\otimes V_{n} = \bigoplus_{k=0}^{n}V_{m+n-2k} $$ for $m\geq n$.
I want to derive a similar formula for $\mathfrak{sl}(3, \mathbb{C})$. In this case, let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{sl}(3, \mathbb{C})$ that consists of diagonal matricies and $L_{i}:\mathfrak{h}\to \mathbb{C}$ be a linear map send $H$ to $H_{ii}$, $(i, i)$ component of $H$. Also, fix a Borel subalgebra $\mathfrak{b}\subset \mathfrak{sl}(3, \mathbb{C})$ as a subalgebra of upper triangular matrices. In this case, positive roots are $\Lambda^{+}=\mathbb{Z}_{\geq 0}\cdot\langle L_{1}, -L_{3}\rangle$ and for each $\lambda \in \Lambda^{+}$, there exists a unique irreducible representation of highest weight $\lambda$.
Now I want to derive a decomposition formula of $V_{\lambda_{1}}\otimes V_{\lambda_{2}}$ for $\lambda_{1}, \lambda_{2}\in \Lambda^{+}$. Here are some examples I computed:
$$ V_{L_{1}}\otimes V_{L_{1}} = V_{2L_{1}}\oplus V_{-L_{3}} \\ V_{L_{1}} \otimes V_{-L_{3}} = V_{L_{1} - L_{3}}\oplus V_{0} \\ V_{2L_{1}}\otimes V_{L_{1}} = V_{3L_{1}} \oplus V_{L_{1}-L_{3}} \\ V_{2L_{1}}\otimes V_{-L_{3}} = V_{2L_{1}- L_{3}} \oplus V_{L_{1}} $$
But I can't find a general formula for $V_{mL_{1}}\otimes V_{-nL_{3}}$ or $V_{\lambda_{1}}\otimes V_{\lambda_{3}}$.