Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $\mathcal{O}$ be the BGG category for $\mathfrak{g}$. It is well-known that the set of irreducible objects are parametrized by $\mathfrak{h}^*$, say it is $\{L(\lambda)| \lambda \in \mathfrak{h}^*\}$.
${\bf Question}:$ Is there any example of $i>1$, $\mathfrak{g}$ and $L(\lambda)$ such that the higher (self)-extension $\text{Ext}^i_{\mathcal{O}}(L(\lambda), L(\lambda))$ is non-zero? Thanks very much in advance!
$Remark$: It is well-known that $\text{Ext}^1_{\mathcal{O}}(L(\lambda), L(\lambda)) =0$ for all $\lambda \in \mathfrak{h}^*$.
This happens already for $\mathfrak{sl}_2$: The only non-semisimple block (up to equivalence) of BGG category $\mathfrak{sl}_2$ has two simple modules, $L$ and $S$ with $\operatorname{dim} \operatorname{Ext}^1(L,S)=\operatorname{dim}\operatorname{Ext}^1(S,L)=\operatorname{dim}\operatorname{Ext}^2(S,S)=1$. In this case, the category has global dimension $2$, so no extensions for $i\geq 2$ appear.