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 $u_\alpha:\mathbb{G}_a \rightarrow G$ such that

(i) $t u_\alpha(c) t^{-1} = u_\alpha( \alpha(t) c)$ for all $c \in k$ 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 Brian Conrad's book 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 difining $u_\alpha(x) = \varphi_\alpha( \begin{pmatrix} 1 & x \\\\ 1 & 0 \end{pmatrix})$ and $u_{-\alpha}(x) = \varphi_\alpha( \begin{pmatrix} 1 & 0 \\\\ x & 1 \end{pmatrix})$ 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 Conrad 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 $p_\alpha: \mathbb{G}_a \rightarrow G$ 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.