  A simple binary matroid is called $I_4$-free if none of its rank-4 flats are
independent sets. These objects can be equivalently defined as the sets $E$ of
points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any
four-dimensional flat $F$. We prove a decomposition theorem that exactly
determines the structure of all $I_4$-free and triangle-free matroids. In
particular, our theorem implies that the $I_4$-free and triangle-free matroids
have critical number at most $2$.
