I'm reading an article which I cannot understand a paragraph very well.

$T$ is a maximal torus of $SU(k+1)$ acting linearly on $\mathbb{C}^{k+1}$. And here is what is written that I cannot fully understand:

Note that the normalizer $N=N(T)$ of $T$ in $O(2k+2)$ normalizes the centralizer $C(T)$ of $T$ in $O(2k+2)$. Assume $k\geq 2$. Since $C(T)$ is a maximal torus of $U(k+1)$ which is also a maximal torus of $O(2k+2)$, we see that $C(T)$ is the identity component of the normalizer $N$. Moreover, $N$ is generated by $C(T)$ , the complex conjugation $c$ and the symmetric group $S_{k+1}$ of permutations of complex coordinates.

First of all, I cannot see why $C(T)$ should be a maximal torus of $U(k+1)$. I can see why it should be connect, since it is the union of maximal tori in $U(k+1)$ containing $T$. My supervisor also said that it is because a element in $C(T)$ lies on $N(T)$, and we can assume, using complex conjugation $c$, that every element $n\in N(T)$ is complex-linear on $\mathbb{C}^{k+1}$ (and I cannot see why). And that, up to permutation, $n$ preserves the decomposition $\mathbb{C}^{k+1}=\mathbb{C}\oplus\cdots\oplus \mathbb{C}$ (which I also cannot see why). And furhter he said that such an element must Lie on the maximal torus $S^1\times\cdots\times S^1\subset U(K+1)$; factorization compatible with the decomposition $\mathbb{C}^{k+1}=\mathbb{C}\oplus\cdots\oplus \mathbb{C}$ (and, again, I cannot see why).

Given this, why should $C(T)$ be the indentity component of $N(T)$?