I am not sure if this is the right place to ask but since I don't get many answers on this kind of questions on SE I might as well try it here. I also posted this question on SE.
I have the following problem and would be grateful if somebody could point me to a reference or give a brief explanation:
I have a semisimple algebraic group $G$ (you can assume in characteristic 0 over an algebraically closed field or even $\mathbb{C}$ if it helps) and an automorphism $\sigma:G\to G$ of order 2. Denote with $H$ the subgroup of $G$ fixed by $\sigma$ and let $N(H)$ be its normalizer in $G$.
The paper that I'm reading makes the following claim:
The subgroup $K$ of $Aut(\mathfrak{g})$ generated by $Ad(N(H))$ and $d\sigma$ is reductive.
They assume that it's obvious that $Ad(N(H))$ is reductive and now since $Ad(N(H))$ has at most index 2 in $K$ we have that $K$ is reductive. Both statements are not quite clear to me right now and I couldn't find anything in the standard literature (I have Borels book and Tauvel/Yu's book on algebraic groups).
Any pointers would be appreciated.