  An attempt is made to understand the root spaces of Kac Moody algebras of
hyperbolic type, and in particular $E_{10}$, in terms of a DDF construction
appropriate to a subcritical compactified bosonic string. While the level-one
root spaces can be completely characterized in terms of transversal DDF states
(the level-zero elements just span the affine subalgebra), longitudinal DDF
states are shown to appear beyond level one. In contrast to previous treatments
of such algebras, we find it necessary to make use of a rational extension of
the self-dual root lattice as an auxiliary device, and to admit non-summable
operators (in the sense of the vertex algebra formalism). We demonstrate the
utility of the method by completely analyzing a non-trivial level-two root
space, obtaining an explicit and comparatively simple representation for it. We
also emphasize the occurrence of several Virasoro algebras, whose interrelation
is expected to be crucial for a better understanding of the complete structure
of the Kac Moody algebra.
