  The notion of a "root base" together with its geometry plays a crucial role
in the theory of finite and affine Lie theory. However, it is known that such a
notion does not exist for the recent generalizations of finite and affine root
systems such as extended affine root systems and affine reflection systems. As
an alternative, we introduce the notion of a "reflectable base", a minimal
subset $\Pi$ of roots such that the non-isotropic part of the root system can
be recovered by reflecting roots of $\Pi$ relative to the hyperplanes
determined by $\Pi$. We give a full characterization of reflectable bases for
tame irreducible affine reflection systems of reduced types, excluding types
$E_{6,7,8}$. As a byproduct of our results, we show that if the root system
under consideration is locally finite then any reflectable base is an integral
base.

Quantum Algebra;Group Theory