  We use the Strong Splitter Theorem to decompose the excluded minor class of
binary matroids with no $E_4$-minor. Using this theorem we can get the
3-decomposers and the extremal internally 4-connected matroids as well as any
other important matroids in the class. The matroid $E_4$ is a self-dual
10-element binary 3-connected matroid that plays a useful role in structural
results. It is a single-element coextension of $P_9$, which is a single-element
extension of the 4-wheel. We show that the extremal matroids in this class are
the binary rank-$r$ spikes $Z_r$, the rank 3 and 4 projective geometries $F_7$
and $PG(3,2)$, respectively, the 17-element internally 4-connected matroid
$R_{17}$, and one 12-element rank-6 matroid. All the other 3-connected members
have $P_9$ or $P_9^*$ as 3-decomposers. As immediate corollaries we get
decomposition results for $EX[P_9^*]$ and $EX[P_9]$ as well as the internally
4-connected members of these classes.

Combinatorics