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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 supplanting my earlier bad answer! :) – paul garrett Jul 18 '11 at 15:11

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