Full symmetric Toda system and vector fields on the group SOn(R)

The paper defines T_f(Ψ) = M(∇f(ΨΛΨ^t))Ψ and proves directly, via a careful Lie-algebraic computation and Lemma 2.4, that for B_+-invariant f,g one has [T_f, T_g] = T_{ {f,g} } where {·,·} is the Lie–Poisson bracket on sl_n (Proposition 3.2; see the definition (3.2), the intermediate commutator formula (3.3), and the final identification using ∇{f,g} in the proof) . The candidate solution gives a geometric proof: it (i) identifies π_*T_{f,Λ} with the Hamiltonian vector field on the SO_n-coadjoint orbit O_Λ via the KKS form, using the decomposition sl_n = so_n ⊕ b^- and B_+-invariance (mirroring the paper’s Lemma 2.4-based cancellations) , and then (ii) invokes the general identity [X_f, X_g] = X_{ {f,g} } on symplectic manifolds to push the Lie bracket through π; injectivity of dπ for simple spectrum and a continuity-in-Λ argument handle multiplicities. A minor imprecision in the model’s phrasing is that O_Λ is a coadjoint orbit (symplectic leaf) for SO_n, not for the Lie–Poisson structure on sl_n per se; this does not affect the argument’s validity. The paper’s proof is independent of this orbit-pushforward route (it explicitly notes kernel issues when the spectrum is not simple and therefore proves the commutator formula directly on SOn) . Net: both reach the same conclusion by different, compatible routes.