Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. Let $N := \operatorname N_G(T)$. My question is: Can I always achieve $N\subseteq KT$? Or differently put, can I find a representative of each $w\in W=N/T$ inside $K$?

My intuition comes from the $G=\operatorname{GL}_n(\mathbb C)$ case where $K$ is the orthogonal group which contains all the permutation matrices which in turn are a set of representatives for the Weyl group with respect to the torus consisting of diagonal matrices.

If this is always possible, I would be happy to have a reference which I can quote for this statement.