On affine Toda field theories related to ${\mathbf{D}}_r$ 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 ${\mathbf{D}}_r$. We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of $\mathfrak{g}$. This is illustrated on the examples ${\mathbf{D}}_{r+1}^{(1)}\to {\mathbf{B}}_r^{(1)}$ and ${\mathbf{D}}_4^{(1)}\to {\mathbf{G}}_2^{(1)}$ using external automorphisms of the corresponding extended Dynkin diagrams.