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Let $\Phi$ be a root system of euclidean space $E$.

Suppose that a subset \Phi'\subset \Phi satisfies \Phi'=-\Phi' and if \alpha,\beta\in\Phi' and $\alpha+\beta\in \Phi$, then \alpha+\beta\in \Phi'.

I want to show that \Phi' is a root system in span($\Phi'$).

Actually, I proved other axioms. The final one is to show the following:

\sigma_{\alpha}(\beta)=\beta-<\beta,\alpha>\alpha\in \Phi', where $<\beta,\alpha>=2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$.

How can I show it?

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Does this not follow from the fact that in $\Phi$ there is an unbroken $\alpha$-string from $\beta$ to $\beta+m\alpha=\sigma_{\alpha}(\beta),$ $m=\langle \beta,\alpha\rangle$? IOW all the vectors $\beta+i\alpha$ with $i=0,1,\ldots,m$ are known to be in $\Phi$, so by induction on $i$ they are also in \Phi' given that \beta,\alpha\in\Phi'.

Edit: I leave it to you to figure out what changes are required in the case $m<0$.

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    (Thx for supplan$t$ing my earlier bad answer! :)2011-07-18