An Unstable Approach to the May-Lawrence Matrix Toda bracket and the \textit{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 $(JS^{m},S^{m}) $ and $(JS^{2m},*)$ localized at 2. After that we provide a generalized $H$-formula for matrix Toda brackets. As an application, we show a new construction of $\ct'\in\pi_{26}(S^{6})$ localized at 2 which improves the construction of $\ct'$ given by \cite{20STEM}.