  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}.
