Recursion operators and hierarchies of $\text{mKdV}$ equations related to the Kac–Moody algebras $D_4^{(1)}$, $D_4^{(2)}$, and $D_4^{(3)}$

We construct three nonequivalent gradings in the algebra $D_4\simeq so(8)$. The first is the standard grading obtained with the Coxeter automorphism $C_1=S_{\alpha_2}S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}$ using its dihedral realization. In the second, we use $C_2=C_1R$, where $R$ is the mirror automorphism. The third is $C_3=S_{\alpha_2}S_{\alpha_1}T$, where $T$ is the external automorphism of order 3. For each of these gradings, we construct a basis in the corresponding linear subspaces $\mathfrak{g}^{(k)}$, the orbits of the Coxeter automorphisms, and the related Lax pairs generating the corresponding modified Korteweg–de Vries (mKdV) hierarchies. We find compact expressions for each of the hierarchies in terms of recursion operators. Finally, we write the first nontrivial mKdV equations and their Hamiltonians in explicit form. For $D_4^{(1)}$, these are in fact two mKdV systems because the exponent 3 has the multiplicity two in this case. Each of these mKdV systems consists of four equations of third order in $\partial_x$. For $D_4^{(2)}$, we have a system of three equations of third order in $\partial_x$. For $D_4^{(3)}$, we have a system of two equations of fifth order in $\partial_x$.