Thank you! I know similar example before, like $H_1=GL_2$ in $G=GL_3$ $H_2=gH_1g^{-1}$ with $g=\begin{pmatrix} 1 & 0 & 1 \\ 0 &1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. I shall mean that whether every solvable group lie in the class... Btw, one can prove that although $H_1 \cap H_2$ may not be reductive, $H_1 \cap gH_2g^{-1}$ is indeed reductive for some $g$.

In moduli problems, people use such criterions to check the moduli functor is smooth, proper etc.

Thank you! I have a simple question: how do we get $\beta : H^{2n-1}(X) \rightarrow H^1(Pic(X)^0_{red}) $ using the Poincare divisor $D$ as in 2A1 (ii)?

@SashaP Thank you, this is a good application of gcd thm in Weil II, which can also be used to show integral etale cohomology is torsion free for large $\ell$.