Relations in the 24-th homotopy groups of spheres

The main purpose of this note is to give a proof of the fact that the Toda brackets \ $\langle\bar{\nu},\sigma,\bar{\nu}\rangle$ and $\langle\nu,\eta, \bar{\sigma}\rangle$ are not trivial. This is an affirmative answer of M.~Mahowald's Conjecture (J. Mukai, Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs \textbf{13}(2008), 355--383). The second purpose is to determine the relations including $\bar{\nu}_6\omega_{14}$ in $\pi^6_{30}$ and $\bar{\nu}_7\omega_{15}$ in $\pi^7_{31}$. To this end, we provide relations between the Toda bracket and the $J$-homomorphism, and between the Toda bracket and the generalized $P$-homomorphism.