The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and triangular $B$ matrices and come with a "Weyl group" $N/(B\cap N)$, that has to be a Coxeter group.
Conversely, Tits showed in 1974 that a group with a spherical BN-pair of rank at least 3 is of "Lie type". (Ironically, the general group identification needed in the classification for lowest rank 3 was the "quasi-thin" case solved last by Aschbacher).
Now the existence of a rank 1 BN-pair for a group $G$ is equivalent to the existence of a doubly-transitive action of $G$ on a set $X$ (which can be taken to be $G/B$). This implies that also the sporadic simple Mathieu groups have a BN-pair (of rank 1).
Now my question: The construction of the Monster group uses also a rank 1 "BN-construction" that is not proper: The "triality" element in the Weyl group $S_3$ takes the (non-normalizing) role the transpositions ought to have. Sometimes the term "weak BN-pair" is mentioned. However, I could not find a proper definition. Can one briefly explain this concept and how it is related to the usual BN-pairs? Is it as generic or rather ad-hoc? Can one suggest good introductory literature?
EDIT: Found e.g. in appendix F of Aschbachers "Classification of Quasithin groups"