Where does non-reduced root system come up?

Question

A reduced root system $R$ (over $\mathbb{R}$) is one which satisfies the condition that $\mathbb{R}\alpha \cap R $ consists of only $\alpha$ and $-\alpha$ for every root $\alpha$ (following Bourbaki's definition of root system). I know that the reduced root systems come up from the root space decomposition of a semisimple lie algebra. Are there subjects where the non-reduced root systems also make a natural appearance?

Answer 1

More precisely you should write that reduced root systems come up in the classification of split semisimple Lie algebras. If your ground field $k$ is not algebraically closed, a semisimple Lie algebra typically has two root systems attached to: Its "absolute" root system, which is the one you get after scalar extension to an algebraic closure, and which by the classical theory is always reduced, and its $k$-rational (or relative) roots, which are the roots for a maximal split toral subalgebra (which is no longer necessarily maximal toral). Cf. here and here.

These $k$-rational roots in a certain sense are a quotient of the absolute root system; they can be empty, or form a root system which possibly is non-reduced.

For the base field $k=\mathbb R$, I gave a list of simple Lie algebras where the relative roots form a system of type $BC$ in Examples of Lie algebras of the $BC$ root system type.