Problem 9.7 - Lie Algebras - Humphreys

Question

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must this coincide with $E'\cap\Phi$?).

More generally, Let $\Phi'$ be a nonempty subset of $\Phi$ such that $\Phi'=-\Phi'$, and such that $\alpha,\beta\in\Phi'$, $\alpha+\beta\in\Phi$ implies $\alpha+\beta\in\Phi'$. Prove that $\Phi'$ is a root system in the subspace of E it spans.

Answer 1

I hope this could be useful for you and I'm not saying something wrong.

I don't use root systems by a lot of time, then I'm going to recall some of their properties solving you execise.

Definition $\Phi \subset E$ is a root system for $E$ if

1. $\Phi$ generates $E$ (in our case this is by definition)
2. $\alpha$ and $-\alpha$ are the only "$\alpha$"-multiples in $\Phi$ (and this is trivial for $\Phi \cap E'$ because $E'$ is a subspace of $E$)
3. $\forall \gamma, \; \delta \in \Phi$ we have $\langle \gamma, \; \delta \rangle \in \mathbb{Z} $ (this holds for the elements of our $\Phi$ and then holds for the elements of its subset $\Phi \cap E'$)
4. If $\gamma \in \Phi$ and $\sigma_{\gamma}$ is the reflection by the iperplane generated by $\gamma$, then $\sigma_{\gamma}(\Phi)=\Phi$

The hardest part of you question is obviously $4.$.

To solve it, we have to ask ourself the following question: How the generic reflection $\sigma_{\gamma}$ acts on the generic root $\delta \in \Phi$?

Well, $\sigma_{\gamma}(\delta)= \delta - \langle \delta, \; \gamma \rangle \gamma $

I think now is easy to conclude...