On affine Toda field theories related to D_r algebras and their real Hamiltonian forms

The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series D_r. We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of 𝔤. This is illustrated on the examples D_r+1^(1)→ B_r^(1) and D_4^(1)→ G_2^(1) using external automorphisms of the corresponding extended Dynkin diagrams.