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 $<\bar{\nu },\sigma ,\bar{\nu }>$ and $<\nu ,\eta ,\bar{\sigma }>$ 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 ${\pi }_{31}^7$.