  We explain how to achieve the traceless gauge for the spatial part of the
spin connection in the framework of the recently proposed correspondence
between the (appropriately truncated) bosonic sectors of maximal supergravities
and the `geodesic' sigma-model over E10/K(E10) at low levels. After making this
gauge choice, the residual symmetries on both sides of this correspondence
match precisely. The gauge choice also allows us to give a physical
interpretation to the multiplicity of certain primitive affine null roots of
E10.
