Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u_\alpha) _{\alpha \in \Phi(G,T)}$ of immersions $u _\alpha:\mathbf{G}_a \rightarrow G$ such that

(i) $t u_\alpha(c) t^{-1} = u_\alpha( \alpha(t) c)$ for all $c \in \mathbf{G}_a$ and $t \in T$,

(ii) $n_\alpha := u_\alpha(1) u_{-\alpha}(-1) u_\alpha(1)$ lies in $\mathrm{N}_G(T) \setminus T$,

(iii) $u_\alpha(x) u_{-\alpha}(-x^{-1}) u_\alpha(x) = \alpha^\vee(x) n_\alpha$ for all $x \in k^\times$,

a *realization* of (G,T) (or $\Phi(G,T)$) in $G$. We then have $\mathrm{Im}(u_\alpha) = U_\alpha$.

In the book of Conrad-Gabber-Prasad on pseudo-reductive groups a *pinning* of $G$ is defined as a tuple $(T,\Phi^+,(\varphi_\alpha)_ {\alpha \in \Delta})$ where $T$ is a maximal torus, $\Phi^+$ is a positive system for $\Phi(G,T)$, $\Delta$ is the corresponding basis and $\varphi _{\alpha}: (\mathrm{SL} _2, \mathrm{SL} _2 \cap \mathrm{D} _2) \rightarrow (G _\alpha, G _\alpha \cap T)$ are central isogenies such that $\varphi _\alpha( \mathrm{diag}(x,x^{-1}) ) = \alpha^\vee(x)$ for all $x \in k^\times$, where $G _\alpha = \langle U _\alpha,U _{-\alpha} \rangle$.

My question is: are these two notions somehow equivalent? If a pinning is given, then by defining

$u_\alpha(x) = \varphi_\alpha\begin{pmatrix} 1 & x \\\\ 1 & 0 \end{pmatrix}$, $u_{-\alpha}(x) = \varphi_\alpha\begin{pmatrix} 1 & 0 \\\\ x & 1 \end{pmatrix}$

I get closed immersions satisfying the properties above, but unfortunately, as I have $\varphi_\alpha$ only for $\alpha \in \Delta$ this does not yet define a realization. How can I define the $u_\alpha$ for $\alpha \notin \Delta \cup -\Delta$? What about the other direction?

Moreover (as C-G-P also mentions) in SGA3, exposé XXIII, there is defined the notion of *épinglages* and Conrad mentions that these carry the same information as the pinnings above. Can somebody make this precise? Moreover in SGA, it is mentioned that an épinglage induces monomorphisms $p_\alpha: \mathbf{G}_a \rightarrow G$ for $\alpha \in \Delta \cup -\Delta$. I suspect that these are the morphisms I defined above, but again, can I get a realization from this?

A further problem is the following: For a given realization and a total order on $\Phi(G,T)$ Springer defines *structure constants* which appear in the expression of the commutator $\lbrack u_\alpha(x), u_\beta(y) \rbrack $ in terms of $u_\gamma$ for linearly independent $\alpha, \beta \in \Phi$. Springer shows that for root systems NOT of type $G_2$ a realization with integral structure constants exist. Demazure also calculates these commutators in SGA3, exposé XXII, for the $p_\alpha$ mentioned above in case of rank 2 root systems. Here, I was surprised that the structure constants seem to be *independent* of the pinning chosen. Is this now a rank 2 phenomenon that is also true for realizations or does this mean that pinnings/épinglages are more restrictive than realizations?

I hope, somebody can help me here.

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