a subset of a root system is root system if it satisfies a property

Question

Let $\Phi$ be a root system in Euclidean space $E_n$.

Let $\Phi'$ be a subset of $\Phi$ with following properties:

(1) $\Phi'=-\Phi'$.

(2) If $\alpha,\beta\in\Phi'$ and $\alpha+\beta\in \Phi$ then $\alpha+\beta\in \Phi'$.

(An exercise in Humphrey's Lie algebra:) Prove that $\Phi'$ is a root system in subspace of $E_n$ spanned by it.

Here, we essentially have to prove that for every $\alpha\in\Phi'$ , the reflecction $\sigma_{\alpha}$ leaves $\Phi'$ invariant. But I am not getting any direction to prove it. How do we prove this? any hint?

I did only this much: let $\alpha,\beta\in \Phi'$ and we have to show that $\sigma_{\alpha}(\beta)\in \Phi'$.

Suppose $\sigma_{\alpha}(\beta)$ is not in $\Phi'$. Then

(i) $\alpha$ is not orthogonal to $\beta$ (because reflection $\sigma_{\alpha}$ is identity on orthogonal complement of $\alpha$.)

(ii) $\sigma_{\alpha}(\beta)\neq \pm\alpha$ (since we assumed that $\sigma_{\alpha}(\beta)\notin \Phi')$

Thus, angle between $\alpha$ and $\beta$ is neither $0$, not $\pi/2$; replacing $\alpha$ by $-\alpha$ we can assume that angle between $\alpha$ and $\beta$ is strictly between $0$ and $\pi/2$; thus $\alpha-\beta$ is a root in $\Phi$. I couldn't proceed further.

(Reflection $\sigma_{\alpha}$ means reflection in the hyperplane orthogonal to vector $\alpha$.)

Answer 1

Hint. Observe that $\sigma_\alpha(\beta)$ is of the form

$\beta + k\alpha$

for some integer $k$.

So can you show, with (1) and (2), that:

If $k \in \mathbb{Z}$ and $\alpha, \beta \in \Phi'$ with $\beta + k \alpha \in \Phi$, then $\beta + k\alpha \in \Phi'$.

This can be done e.g. by induction backward and forward from $k = 0$; you might need that the possible $k$'s here (in the "$\alpha$-chain of $\beta$" or "$\alpha$-root string through $\beta$") are of the form $-p, -p+1, ..., q-1, q$ -- maybe that has been covered in the book before the exercise.