  Recently, the concept of a nonlinear sigma-model over a coset space G/H was
generalized to the case where the group G is an infinite-dimensional Kac-Moody
group, and H its (formal) `maximal compact subgroup'. Here, we study in detail
the one-dimensional (geodesic) sigma-model with G = E10 and H=KE10. We
re-examine the construction of this sigma-model and its relation to the bosonic
sector of eleven-dimensional supergravity, up to height 30, by using a new
formulation of the equations of motion. Specifically, we make systematic use of
KE10-orthonormal local frames, in the sense that we decompose the `velocity' on
E10/KE10 in terms of objects which are representations of the compact subgroup
KE10. This new perspective may help in extending the correspondence between the
E10/KE10 sigma-model and supergravity beyond the level currently checked.
