The weyl group preserves roots and root multiplicities

For $D=\mathrm{ad}{e}_i$, F2 gives nilpotence on each $e_j$; the relations give $D{f}_j={\delta }_{ij}{h}_i$, $D^2{f}_i=-2e_i$, $D^3{f}_i=0$, and $D^{2}h=0$. For any derivation, $D^m[x,y]={\sum }_{k=0}^m\left(\genfrac{}{}{0px}{}{m}{k}\right)[D^{k}x,D^{m-k}y]$, proved by induction and Pascal addition. Hence nilpotence on generators propagates to every finite bracket word and every finite sum. The same holds for $\mathrm{ad}{f}_i$. Their pointwise finite exponentials preserve brackets by the binomial identity, with inverses obtained by negating the derivation.

Set $T_i=\exp (\mathrm{ad}{f}_i)\exp (-\mathrm{ad}{e}_i)\exp (\mathrm{ad}{f}_i)$. On the triple, the finite expansions use $\exp (\mathrm{ad}{f}_i)e_i=e_i-h_i-f_i$, $\exp (\mathrm{ad}{f}_i)h_i=h_i+2f_i$, and $\exp (-\mathrm{ad}{e}_i)f_i=f_i-h_i-e_i$. Substitution gives $T_i{e}_i=-f_i$, $T_i{f}_i=-e_i$, and $T_i{h}_i=-h_i$. It fixes $\ker {\alpha }_i$ in $\mathfrak{h}$. Writing $h=(h-{\alpha }_i(h)h_i\mathrm{/}2)+{\alpha }_i(h)h_i\mathrm{/}2$ gives $T_{i}h=h-{\alpha }_i(h)h_i$.

For $x\in {\mathfrak{g}}_{\beta }$, $[h,T_{i}x]=T_i[T_i^{-1}h,x]=\beta (s_{i}h)T_{i}x=(s_i\beta )(h)T_{i}x$. The inverse automorphism gives a bijection between these spaces. Products of $T_i$ implement each word in the generators of $W$, proving root and multiplicity invariance. Only the induced weight action, not independence of the lift from a word, is required.

If $DA$ is symmetric, extend $({\alpha }_i,{\alpha }_j)=d_i{a}_{ij}$ bilinearly. For $\lambda$ in the root span, $({\alpha }_i,\lambda )=d_i\lambda (h_i)$ and $({\alpha }_i,{\alpha }_i)=2d_i$. Expanding $(\lambda -\lambda (h_i){\alpha }_i,\mu -\mu (h_i){\alpha }_i)$ cancels the two cross terms against $2d_i\lambda (h_i)\mu (h_i)$, leaving $(\lambda ,\mu )$. This computation allows a degenerate form.