Leibniz rule on higher pages of unstable spectral sequences

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": $d^r(a\odot σb)=\pm d^ra\odot b+a\odot d^rσb,$ where $σ$ is the suspension homomorphism.

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