Rheological and structural properties were examined for a 30 wt % solution of a 1,4-butadiene-styrene-1,4-butadiene (BSB) triblock copolymer (M-S = 108 x 10(3), M-B = 13.3 x 10(3) for each B block) in an S-selective solvent, dibutyl phthalate (DBP). At equilibrium at 25 degreesC, a bee lattice of the unsolved, soft (rubbery) B domains connected by the middle S blocks (a lattice-type network) was formed to exhibit the elastic behavior against small strains. This lattice-type network was disrupted under steady shear to lose its elasticity, and the strongest disruption occurred at an intermediate shear rate (gamma) over dot close to a frequency of B/S concentration fluctuation. This disruption behavior was similar to that of a BS/DBP diblock micellar lattice system. However, for the BSB/DBP system, an order of magnitude increase of (gamma) over dot resulted in a minor change (by a factor <30%) of a time t(r)(infinity) required for the full recovery of the elasticity during a quiescent rest after the preshear. This result, quantitatively different from that seen for the BS/DBP diblock system, was related to the bridge-type configuration of the middle S blocks of BSB: Under the steady shear, a large fraction of the bridges in the flowing defect region would be converted into loops. Thus, the re-formation of the bridges in this region (a process absent in the BS/DBP system) should be required for the full recovery of the elasticity of the BSB/DBP system. This bridge re-formation, leading to a recovery of the order of the B domain arrangement and allowing the coexisting loops to exhibit the elasticity, should be accompanied by a transient mixing of the B and S block and thus gave the very weakly (gamma) over dot-dependent t(r)(infinity) determined by the mixing enthalpy.