Decays of B¯ 0 s and B¯ 0 mesons into J=ψ πþπ−πþπ− final states, produced in pp collisions at the LHC, are investigated using data corresponding to an integrated luminosity of 3 fb−1 collected with the LHCb detector. B¯ 0 ðsÞ → J=ψf1ð1285Þ decays are seen for the first time, and the branching fractions are measured. Using these rates, the f1ð1285Þ mixing angle between strange and nonstrange components of its wave function in the qq¯ structure model is determined to be ð24:0þ3.1þ0.6 −2.6−0.8 Þ°. Implications on the possible tetraquark nature of the f1ð1285Þ are discussed