On some components of $L(\rho)\otimes L(\rho)$ associated with rooted trees for symmetrizable Kac-Moody algebras

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra over $\mathbb{C}$ and let $L(\rho)$ be the irreducible integrable $\mathfrak{g}$-module with highest weight $\rho$. Let $I$ be a subgraph of the Dynkin diagram of $\mathfrak{g}$ which has only simple bonds and no cycle of length $\geq 3$. For every subset $D$ of $I$, denote by $\beta_D$ the sum of the simple roots corresponding to $D$. To every $D \subset I$ such that $\lambda_{D,I} = 2\rho - \beta_I - \beta_D$ is dominant, we associate certain elements $\pi_{D,I}$ of weight $\lambda_{D,I} {-} \rho$ in the crystal $B(\rho)$, which depend on the choice of a root vertex in each connected component of $I$. Then we prove that our elements are $\rho$-dominant
elements of $B(\rho)$, hence provide new families of components
of the tensor product $L(\rho)\otimes L(\rho)$.