Root of a root system
= Root of a root system
= Roots of a root system
{synonym}
A root is an individual nonzero vector in a <root system>. For a complex <semisimple Lie algebra>, it is a nonzero <weight> of the <Adjoint representation> restricted to a <Cartan subalgebra>; its <root space> consists of vectors $x$ satisfying $[h,x]=\alpha(h)x$ for all $h$ in that subalgebra.