A natural composition o on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for the p-lower central series spectral sequence of a simplicial group. It is proved that the rth differential satisfies a 'Leibniz rule with suspension': d^r(a o σ b) = ±d ^r a o b + a o d ^r σ b, where σ is the suspension homomorphism.