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Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, and for each root subgroup $U_{\alpha}$ of $U$, let $u_{\alpha}: \mathbb R \rightarrow U_{\alpha}$ be an isomorphism.

For all $\alpha, \beta \in \Phi^+$, there exist unique real numbers $C_{\alpha,\beta,i,j}$ (depending on the $u_{\alpha}$ and the ordering) such that for all $x, y \in \mathbb R$,

$$u_{\alpha}(x) u_{\beta}(y) u_{\alpha}(x)^{-1} = u_{\beta}(y) \prod\limits_{\substack{i,j>0\\ i\alpha + j \beta \in \Phi^+}} u_{i\alpha+j\beta}(C_{\alpha,\beta,i,j}x^iy^j)$$

I want to work out some examples with unipotent groups of exceptional semisimple groups, and am looking for table of structure constants for the root system G2. Does anyone know a reference where an ordering on the roots is chosen and these constants are written down?

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    $\begingroup$ If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them. $\endgroup$ – Ben McKay Feb 10 at 7:02
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    $\begingroup$ Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature. $\endgroup$ – Gro-Tsen Feb 10 at 9:18
  • $\begingroup$ Correction: $U$ is the unipotent radical of $B$ in this formulation. $\endgroup$ – Jim Humphreys Feb 10 at 15:16
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"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of $C_{\alpha\beta11}$ are listed. An explicit form of commutator formulas inside $U^+$ is given in Table IV of "Chevalley groups over commutative rings: I. Elementary calculations" by N. Vavilov and E. Plotkin, see the picture below.

enter image description here

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  • $\begingroup$ Thanks! So just to be clear $ij$ means $i\alpha + j \beta$, where $\alpha$ is the short root, right? $\endgroup$ – D_S Feb 10 at 15:58
  • $\begingroup$ @D_S Indeed, as indicated by the presence of $3\alpha+2\beta$. $\endgroup$ – Andrei Smolensky Feb 10 at 16:17
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SGA III, Expose XXIII, Section 3.4.

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Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)

Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.

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