  This paper connects the quadrirational Yang-Baxter maps, which are
two-dimensional integrable discrete systems of KdV type, and the elliptic
Cremona system, which is a higher analogue of discrete Painlev\'e equations
associated with $\tilde{E}_8$ symmetry. This is a natural connection between
integrable systems in different dimensions that is outside of the usual
paradigm of reductions. Our approach is based on formulation of both systems in
terms of birational Coxeter groups.
