  Extended affine root systems appear as the root systems of extended affine
Lie algebras. A subclass of extended affine root systems, whose elements are
called ``minimal" turns out to be of special interest mostly because of the
geometric properties of their Weyl groups; they possess the so-called
``presentation by conjugation". In this work, we characterize minimal extended
affine root systems in terms of ``minimal reflectable bases" which resembles
the concept of the ``base" for finite and affine root systems. As an
application, we construct elliptic Lie algebras by means of Serre's type
generators and relations.
