I'm studying root systems and coming up with an observation:

Let $\Phi$ be an irreducible root system and $\Phi^+$ be a positive root system. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ be the corresponding base. For $\beta, \gamma \in \Phi^+$, $\alpha_1, \alpha_2 \in \Delta$ with the properties that $\beta=\gamma +\alpha_1+\alpha_2$, $\gamma +\alpha_1 \in \Phi^+, \gamma +\alpha_2\in \Phi^+, \alpha_1+ \alpha_2 \notin \Phi^+$ then

Statement: There exists $\alpha_3 \in \Delta\setminus\{\alpha_1,\alpha_2\}$ such that either $\gamma +\alpha_1-\alpha_3 \in \Phi^+$ or $\gamma +\alpha_2-\alpha_3 \in \Phi^+$.

The motivation is to investigate the relationship between a positive positive root and other lower positive roots. That is to understand the so-called Hasse diagram root systems (for instance see definition of Hasse diagram in here)

Any help would be much appreciated.