Factorization of loop algebras over $\mathrm{so}(4)$ and integrable nonlinear differential equations

We consider factoring subalgebras for loop algebras over $\mathrm{so}(4)$. Given a factoring subalgebra, we find (in terms of coefficients of commutator relations) an explicit form of (1) the corresponding system of the chiral field equation type, (2) the corresponding two-spin model of the Landau–Lifshitz equation, and (3) the corresponding Hamiltonian system of ordinary differential equations with homogeneous quadratic Hamiltonian and linear $\mathrm{so}(4)$-Poisson brackets.