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Let ( G, B, N ) be a group with an irreducible spherical BN-pair of rank at least 2, and let  U  be a nilpotent normal subgroup of  B  such that  B  =  U  ( B  ⋂  N ). We show that  U  is unique with respect to  B . As a corollary, we obtain a complete classification of all irreducible spherical split BN-pairs of rank at least two.

Split BN-Pairs of Rank at Least 2 and the Uniqueness of Splittings