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In this paper, we give an unstable approach of the May-Lawrence matrix Todabracket, which becomes a useful tool for the theory of determinations ofunstable homotopy groups. Then, we give a generalization of the classicalisomorphisms between homotopy groups of $(JS^{m},S^{m}) $ and $(JS^{2m},*)$localized at 2. After that we provide a generalized $H$-formula for matrix Todabrackets. 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}.

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