An Unstable Approach to the May-Lawrence Matrix Toda bracket and the $\mathit{\text{2}}$nd James-Hopf Invariant

In this paper, we give an unstable approach of the May-Lawrence matrix Toda bracket, which becomes a useful tool for the theory of determinations of unstable homotopy groups. Then, we give a generalization of the classical isomorphisms between homotopy groups of $(J{S}^m,S^m)$ and $(J{S}^{2m},\ast )$ localized at 2. After that we provide a generalized $H$-formula for matrix Toda brackets. As an application, we show a new construction of ${\text{ct}}^{\mathrm{\prime }}\in {\pi }_{26}(S^6)$ localized at 2 which improves the construction of ${\text{ct}}^{\mathrm{\prime }}$ given by \cite{20STEM}.