This is part of proposition 1.20 in Michel and Digne's Finite Groups of Lie Type.
A closed subgroup $P$ of $G$ which contains $T$ (maximal torus) is parabolic iff for any root $\alpha\in\Phi$, either $U_\alpha\subset P$ or $U_{-\alpha}\subset P$. (These are root subgroups.)
They let $B'$ be a Borel of $P$, and $B$ a Borel of $G$ containing $B$, and aim to show $B'=B$. Let $\alpha>0$. They show if $U_\alpha\subseteq P$, then $U_\alpha\subseteq B'$. I get that. Assuming now $U_\alpha\not\subset P$, I don't get the following:
Otherwise $U_{-\alpha}\subset P$. If $U_\alpha\not\subset B'$, then $U_\alpha\not\subset R_u(P)$, so $U_\alpha$ maps isomorphically to a root subgroup relative to the image of $T$, of the reductive group $P/R_u(P)$. Then $-\alpha$ is also a root of this group. Let $n$ be a representative in $P$ of the reflection $s_{-\alpha}$ in the Weyl group of $P/R_u(P)$; the element $n$ conjugates $T$ to another torus of the group $TR_u(P)$, so as all maximal tori in this group are conjugate, $n$ can be changed to another represenative $n'$ which normalizes $T$. We note then that on $^{n'}U_{-\alpha}$ the group $T$ acts by $s_{-\alpha}(-\alpha)=\alpha$ which contradicts $U_\alpha\not\subset P$.
I can't follow them after the bolded part above. What explicitly is the change of $n$ to $n'$, what does it mean that $T$ acts by $\alpha$ on $^{n'}U_{-\alpha}$, and how does that contradict $U_\alpha\not\subset P$?