Extension Data and Their Lie Algebras

We state the notion of an extension datum of a root system. This concept generalizes the root system extended by an abelian group. We classify them, and show that there is a one-to-one correspondence between (reduced) root systems extended by an abelian group and (reduced) tame irreducible extension data. We also show that an extension datum, modulo some isotropic roots, is the union of a finite number of tame irreducible extension data.