Let V be a vector space over an algebraically closed field. 


Let $\{H\_i\}\_{i \in I}$ be a collection of closed connected subgroups of $\operatorname{GL}(V)$ (wrt. Zariski topology). It is a basic result in algebraic groups that the subgroup $G$ generated by the $H\_i$ is a closed connected subgroup of $\operatorname{GL}(V)$. In fact, we have $G = H\_{i\_1} \cdots H\_{i\_n}$ for some $i\_1, \ldots i\_n \in I$. (For example 7.5 "Linear Algebraic Groups" by Humphreys).


Is the Lie algebra of $G$ generated by the Lie algebras of the $H\_i$? 



Lie algebra of an algebraic group generated by connected subgroups
Let $p=\mathrm{char}(k)>0$. The vector group $G=\mathbf{G}\_a^2$ is generated by $H\_1=\{(t,0)|t\in k\}$ and $H\_2=\{(t,t^p)\mid t\in k\}$ but $\mathrm{Lie}H\_1=\mathrm{Lie}H\_2=k\oplus 0$.


