Left-invariant distributions and metric Hamiltonians on ${\rm SL}(n,{\mathbb R})$ induced by its Killing form

From the classical theory of Lie algebras, it is well-known that the bilinear form $B(X,Y)={\rm tr}(XY)$ defines a non-degenerate scalar product on the simple Lie algebra ${\mathfrak{sl}}(n,{\mathbb R})$. Diagonalizing the Gram matrix $Gr$ associated with this scalar product we find a basis of ${\mathfrak{sl}}(n,{\mathbb R})$ of eigenvectors of $Gr$ which produces a family of bracket generating distributions on ${\rm SL}(n,{\mathbb R})$. Consequently, the bilinear form $B$ defines sub-pseudo-Riemannian structures on these distributions. Each of these geometric structures naturally carries a metric quadratic Hamiltonian. In the present paper, we construct in detail these manifolds, study Poisson-commutation relations between different Hamiltonians, and present some explicit solutions of the corresponding Hamiltonian system for $n=2$.