Let $(G,B,N,S)$ be a Tits system. Assume the system is saturated, which is to say that if we define
$$H = \bigcap\limits_{n \in N} nBn^{-1}$$
then $H \subseteq N$. To an affine root system whose Weyl group $(W,S)$ is isomorphic to that of the Tits system, we can associate a Bruhat-Tits building with a standard apartment. The claim that $N$ is the stabilizer of the standard apartment is equivalent to the following statement which I can trying to understand the proof of:
Let $g \in G$, and suppose for every $n \in N$, there exists $n' \in N$ such that $gnBn^{-1}g^{-1} = n'Bn'^{-1}$. Then $g \in N$.
Taking $n = 1$, and using the fact that parabolic subgroups are self-normalizing, we see that there exists $n'' \in N, b \in B$ such that $g = n''b$. Again using the self-normalizing property, we have that for every $n \in N$, there exists an $n' \in N$ such that $bn \in n''^{-1}n'B$.
Then I get lost. The next claim is that $n \in n''^{-1}n'H$, and then that $b \in nBn^{-1}$ for every $n \in N$. I don't understand why either of these claims are true.
The reference is from Bruhat-Tits article I, (2.2.5).
