Let $G$ be a compact connected Lie group with maximal torus $T$. The Weyl group is defined by $$W:=N_G(T)/T.$$ Now, $G$ has a complexification $G_{\Bbb C}$ with maximal torus $T_{\Bbb C}$ which is the complexification of $T$.
Question: Is $W$ naturally isomorphic to the group $W_{\Bbb C}:=N_{G_{\Bbb C}}(T_{\Bbb C})/T_{\Bbb C}$?
Attempt: Since $T\subseteq T_{\Bbb C}$, we have a natural group homomorphism $$N_G(T)/T\longrightarrow N_{G_{\Bbb C}}(T_{\Bbb C})/T_{\Bbb C},\quad gT\longmapsto gT_{\Bbb C}.$$ Moreover, it is easy to see that it is injective. But I am not sure if it is surjective.
Also, $W$ and $W_{\Bbb C}$ both act on ${\frak t}_{\Bbb C}$ by the adjoint representation and it seems natural that these two actions are the same. Is that true?