Abstract
A result of Seymour implies that any 3-connected matroid with a modular 3-point line is binary. We prove a similar characterization for 3-connected matroids with modular 4-point lines. We show that such a matroid is either representable over ${GF}(3)$ or ${\rm GF}(4)$ or has an $F_7$-minor and either an $F_7^-$- or $(F_7^-)^*$-minor.
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