Prenilpotent pairs in the E-10 root lattice

Tits has defined Kac-Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E-10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant) .k(7) as k -> infinity. Our purpose is to motivate alternate approaches to Tits' groups, such as the one in [2].