  We study extensions of the classical Toda lattices at several different
space-time scales. These extensions are from the classical tridiagonal phase
spaces to the phase space of full Hessenberg matrices, referred to as the Full
Kostant-Toda Lattice. Our formulation makes it natural to make further
Lie-theoretic generalizations to dual spaces of Borel Lie algebras. Our study
brings into play factorizations of Loewner-Whitney type in terms of canonical
coordinatizations due to Lusztig. Using these coordinates we formulate precise
conditions for the well-posedness of the dynamics at the different space-time
scales. Along the way we derive a novel, minimal box-ball system for Full Toda
that doesn't involve any capacities or colorings, as well as an extension of
O'Connell's ODEs to Full Toda.
