Quadratic rank-one groups and quadratic Jordan division algebras

A rank-one group X is a group generated by two distinct nilpotent subgroups A and B such that, for each a is an element of A(#), there exists a b is an element of B-# satisfying A(b) = B-a and vice versa. It has been shown that the notions of a rank-one group and of a group with a split BN-pair of rank 1 are equivalent. Hence all algebraic groups of relative rank I and classical groups of Witt index I are rank-one groups. The rank-one group X = < A, B > is said to be quadratic if there exists a ZX-module V satisfying [V, X, X] not equal 0 = [V, A, A]. In the main result of this paper we classify the quadratic rank-one groups, that is, we show that there is a one-to-one correspondence of such groups with special quadratic Jordan division algebras.