Real root spaces are one dimensional sl2 roots

Choose a finite word for $w$ and multiply the automorphisms of F2 to get $T$. Applying $T$ to $[e_i,f_i]=h_i$, $[h_i,e_i]=2e_i$, and $[h_i,f_i]=-2f_i$ gives a triple $e_{\alpha }=T{e}_i$, $h_{\alpha }=T{h}_i$, $f_{\alpha }=T{f}_i$ with those same brackets. Each vector is nonzero; they lie in three distinct weight spaces $\alpha ,0,-\alpha$, hence are independent. Mapping the standard three matrix generators of ${\mathfrak{s}\mathfrak{l}}_2$ to them is a bracket-preserving linear bijection.

F2 and F3 give ${\mathfrak{g}}_{\alpha }=T{\mathfrak{g}}_{{\alpha }_i}=\mathbb{C}{e}_{\alpha }$ and the analogous negative equality. If $c\alpha$ were another root, $T^{-1}$ would send its nonzero space to degree $c{\alpha }_i$, so F3 forces $c=1$ or $c=-1$. The bracket line $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha }]$ is therefore the nonzero line $\mathbb{C}{h}_{\alpha }$, independent of $T$. Evaluation by $\alpha$ is nonzero on this line since $\alpha (h_{\alpha })=2$ from step 1.1. There is exactly one element of this line with that evaluation, proving independence of the normalized coroot.