 The reduced affine root systems were used by Kac and Moody in their work on Kac–Moody algebras.Possibly non-reduced affine root systems were introduced and classified by and (except that both these papers accidentally omitted the Dynkin diagram ).The affine roots systems "A"1 = "B"1 = "B" = "C"1 = "C" are the same, as are the pairs "B"2 = "C"2, "B" = "C", and "A"3 = "D"3

The number of orbits given in the table is the number of orbits of simple roots under the Weyl group.