Let $X = \Omega_0^3S^3$ a connected component of $\Omega^3S^3$. I am interested in explicit construction of spectral sequence converging to odd prime torsion in homotopy groups of $X$.
There is a spectral sequence converging to $p$-torsion in homotopy groups of Kan loops on $S^3$. The first list is given by so called lambda algebra, this is quite an explicit description.
Is it possible to obtain some short and tractable description of the first list of homotopy spectral sequence for the space $X$?