  We study a saturated fusion system F on a finite 2-group S having a Baumann
component based on a dihedral 2-group. Assuming F is 2-perfect with no
nontrivial normal 2-subgroups, and the centralizer of the component is a cyclic
2-group, it is shown that F is uniquely determined as the 2-fusion system of
L_4(q_1) for some q_1 = 3 (mod 4). This should be viewed as a contribution to a
program recently outlined by M. Aschbacher for the classification of simple
fusion systems at the prime 2. The corresponding problem in the component-type
portion of the classification of finite simple groups (the L_2(q), A_7 standard
form problem) was one of the last to be completed, and was ultimately only
resolved in an inductive context with heavy artillery. Thanks primarily to
requiring the component to be Baumann, our main arguments by contrast require
only 2-fusion analysis and transfer. We deduce a companion result in the
category of groups.
