Prenilpotent pairs in E10

Tits has defined Kac-Moody groups for all root systems, over all commutative rings. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his definition explicitly requires knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is futile. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as the seventh power of k, as k approaches infinity. Our purpose is to motivate alternate approaches to Tits' groups.