Let $G$ be a connected reductive group over a field $k$, let $T$ be a maximal torus of $G$, and let $S$ be the maximal $k$-split torus in $T$. Let $M=C_G(S)$. Is $N_M(T)$ connected?
Edit: Suppose that $S$ is maximal among $k$-split tori of $G$. The motivation for this question was to show that the action of $N_M(T)$ on $T$ by conjugation can be realized by elements of $M(F)$. I am mostly interested in the case that $F=\mathbb{R}$ if this assumption helps.
The answer is no!
Consider the group $M/S$. We know that if $G$ is smooth and reductive, then $M$ is smooth reductive. Now, consider the group schemes $(M/S)_{\overline{k}} = M_{\overline{k}}/S_{\overline{k}}$, which is a smooth group scheme.
Consider the subgroup $N_{M/S}(T/S)$. To check if it is connected it is enough to check if $N_{M/S}(T/S)_{\overline{k}}$ is connected by the following lemma : https://stacks.math.columbia.edu/tag/04KV
Let $N' = (N_{M/S}(T/S)_{\overline{k}})_{red}$, and $N = (N_{M}(T)_{\overline{k}})_{red}$. These are smooth group varieties. Thus it is enough to check if $N$ is connected. Now it can be checked that $\pi \otimes_k{\overline{k}}(N) = N'$. Thus we see that it is enough to check if $N'$ is connected.
But $N'$ is not connected since $N'/(T/S)_{\overline{k}}$ is a non-trivial finite scheme(unless weyl group is zero, which is possible if there are no roots, that is $(T/S)_{\overline{k}}$ is trivial, that is $T = S$).