Let $\mathfrak{g}$ be a Kac-Moody Algebra with GCM $A$. Let $\alpha$ and $\beta$ be two roots not necessarily real and $g_\alpha$ and $g_\beta$ be the corresponding weight spaces of dimension $\operatorname{mult} \alpha$ and $\operatorname{mult} \beta$.
My question is: The number $k = \operatorname{mult} \alpha \times \operatorname{mult} \beta$ has 'any' interpretation in Lie theory like $k$ is also the dimension of some weight spaces of some other Lie algebras some way connected to $\mathfrak{g}$ or in some tensor products of $\mathfrak{g}$ like that.
I want to interpret this number $k$ in terms some Lie theory objects. Any suggestion is welcomed.
Thanks for your valuable time.