Sign conventions for a Chevalley basis of a simple complex Lie algebra Let $R$ be the root system of a simple complex Lie algebra $g$ with respect to some Cartan subalgebra $h$.  Chevalley showed there is a basis of $g$ given by the simple coroots {$H_{\alpha_i}=\alpha_i^\vee\in h$} and root vectors $X_\alpha\in g_\alpha$ for each $\alpha\in R$.  This basis has the following properties:
$[H_{\alpha_i},H_{\alpha_j}]=0$
$[H_{\alpha_i},X_\beta]=\beta(H_{\alpha_i})X_\beta$
$[X_{\alpha},X_{-\alpha}]=H_\alpha=\alpha^\vee\in h$
($\ast$)  $[X_\alpha,X_\beta]=\pm(p+1)X_{\alpha+\beta}$, when $\alpha+\beta\in R$ and $p$ is the greatest positive integer such that $\beta-p\alpha\in R$.  Otherwise, if $\alpha+\beta$ is not a root, then the bracket is zero.
References for this can be found in Serre's book on semisimple complex Lie algebras or Humphrey's book or Wikipedia.

Does anybody know a simple way to determine the sign $\pm$ in the fourth property ($\ast$)?

I cannot find a reference and my French is not good, so reading the original works by Chevalley and Tits isn't a viable option.  In particular, I need to find a sign convention that will work for $g$ of type $F_4$.
Thanks so much.
 A: As Florian suggests, there is no canonical choice of structure constants in Chevalley's approach (or any other I'm aware of).   But for the irreducible root systems, especially those of exceptional type, specific sign choices have been made in various papers (probably with some duplication of effort).  One explicit source for type $F_4$ is Table 1 in an old paper by Toshiaki Shoji, published in J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 1-17.   This paper deals with conjugacy classes of Chevalley groups of type $F_4$ over finite fields of odd characteristic.   Though I've never checked the arithmetic, Shoji's papers are usually reliable.   
Over the years I've encountered explicit tables for other root systems but don't have these at hand.   Computer methods have been used by N.A. Vavilov for root systems of type $E$ in one paper (where he notes that signs for $F_4$ can be deduced from those for $E_6$, via folding of the Dynkin diagam).   Here is the full MathSciNet citation:
MR1875718 (2002k:17022) Vavilov, N. A.  Do it yourself structure constants for Lie algebras of types $E_l$.  Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), Vopr. Teor. Predst. Algebr. i Grupp. 8, 60–104, 281; translation in J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1513–1548.
Probably the recent computational work of W.A. de Graaf is relevant too.
A: There is a good discussion of these issues in the paper of A. Cohen, S. Murray and D.E. Taylor, "Computing in groups of Lie type", Math. Comp. 73, Number 247,
1477–1498, (2003), especially section 3 (referring to earlier work, e.g., of Carter). They explain in particular how the signs can be all reduced to so-called "extraspecial pairs", which can be chosen arbitrarily. 
In Magma at least, one can see which extraspecial signs have been chosen using the "ExtraspecialSigns" command.  For instance, one can see using this that GAP and Magma use (or used, I haven't checked the latest versions...) different constants for E_8.
