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 $$and$$ are not trivial. This is an affirmative answer of the second author's Conjecture (Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs 13(2008), 355-383). The second purpose is to show the relation $\bar{\nu}_7\omega_{15}=\nu_7\sigma_{10}\kappa_{17}$ in $\pi^7_{31}$.Comment: 27 page