Let $G$ be a linear algebraic group over field $k$ of characteristic zero. It is well known that the category of finite dimensional $k$--linear representations of $G$ is abelian, and that it is semisimple if and only if $G$ is reductive.

What can be said about the group $G$, if $\mathrm{Ext}^2(V,W)=0$ for all finite dimensional $k$--linear representations $V$ and $W$ of $G$?

One can of course ask the same question with $\mathrm{Ext}^n(V,W)=0$ for any $n\geq 0$. For the moment, I would already be happy to see some examples where a nonzero $\mathrm{Ext}^2$ occurs.