= Root subsystem
{title2=$\Psi\subseteq\Phi$}
= Sub-root system
{synonym}
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.
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