Let $R$ be a root system (reduced, crystallographic) with Weyl Group $W$. Let $(-,-)$ be the usual $W$-invariant scalar porduct on the real vector space spanned by the roots.
Claim. Let $\alpha$ be a long root and let $\gamma$ be a root non-proportional to $\alpha$ (i.e. $\gamma\neq\pm\alpha$). Then we have $$ (\gamma,\alpha^\vee)\in\{-1,0,1\}\,. $$ Here, we denote by $\alpha^\vee$ the root dual to $\alpha$, i.e. $\alpha^\vee=\frac{2\alpha}{(\alpha,\alpha)}$.
To prove this claim, I was thinking to look at the root subsystem spanned by $\alpha$ and $\gamma$. This root subsystem is of rank two. There are only four types of root systems of rank two (namely $\mathsf{A}_1\times\mathsf{A}_1$, $\mathsf{A}_2$, $\mathsf{B}_2=\mathsf{C}_2$ and $\mathsf{G}_2$). We can go through them and check each time that the claim is true in view of the fact that $\alpha$ is long.
Question. Is this reasoning valid? If yes and the claim is indeed true, does anyone know an elegant proof of it which does not involve explicit checking the types? Does anyone know a type-independent proof which does not use the classification of root systems (of rank two or whatever kind of)?
Thank you in advance for reading. (I know the question is elementary in nature but hopefully not completely useless.)
Intuitively, it's helpful to draw a picture: if $\gamma$ is no longer than $\alpha$, then subtracting $k\alpha$ with $|k|>1$ from $\gamma$ will necessarily result in a vector longer than $\alpha$ - but the length of $\gamma$ is preserved under reflection.
Formally, you can do a quick calculation: $|\langle\gamma,\alpha^{\vee}\rangle|=2\frac{|\langle\gamma,\alpha\rangle|}{\langle\alpha,\alpha\rangle}<2\frac{\|\gamma\|\|\alpha\|}{\|\alpha\|^2}\leq 2$ by Cauchy-Schwarz.