On rectangular Toda brackets and Oda's extension problems

This paper tackles $\mathit{\text{N.}}\mathit{\text{Oda}}$'s extension problems for the homotopy groups ${\pi }_{39}(S^6)$, ${\pi }_{40}(S^7)$, and ${\pi }_{41}(S^8)$ localized at 2, the issues having eluded resolution for more than four decades. We introduce a tool for the theory of determinations of unstable homotopy groups, namely, the rectangular Toda bracket, by which we are able to solve the extension problems with respect to these three homotopy groups.