  A natural composition $\odot$ on all pages of the lower central series
spectral sequence for spheres is defined. Moreover, it is defined for $p$-lower
central series spectral sequence of a simplicial group. It is proved that $r$th
differential satisfies a "Leibiz rule with suspension": $d^r(a\odot \sigma
b)=\pm d^ra\odot b+a\odot d^r\sigma b,$ where $\sigma$ is the suspension
homomorphism.
