The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the standard terminology, we call a sub-root system $\Phi^{\prime}\subset \Phi$ *closed* if for any $\alpha$, $\beta\in \Phi^{\prime}$, $\alpha+\beta\in \Phi$ implies $\alpha+\beta\in \Phi^{\prime}$.

The Borel-de Siebenthal theorem classifies all closed sub-root systems of irreducible crystallographic root systems. See also Chapter 12 of Kane's book
*Reflection Groups and Invariant Theory*.

My question is: for a irreducible crystallographic root system $(\Phi, E)$, can we find two closed sub-root systems $\Phi_1$ and $\Phi_2$ such that $\Phi_i \neq \Phi$ for $i=1,2$, and $\Phi_1\cup \Phi_2=\Phi$?

I believe the answer is negative and since we have the classification, we may get a proof through a exhausting all maximal closed sub-root systems. I wonder if we can prove it more theoretically.