Root subsystem ($\Psi \subseteq \Phi$)

A root subsystem of a root system $\Phi$ is a subset that is itself a root system in its real span. In particular, if a subspace $F$ is spanned by roots, $\Phi \cap F$ is a full root subsystem: root reflections in its roots preserve both $\Phi$ and $F$. Reflection-generated subsystems need not contain every ambient root in their span. A pair is a simple root basis only when every subsystem root has integral coordinates of one sign in that pair; merely choosing two independent roots is insufficient.