Designs related through projective and Hopf maps

We verify a construction which, for $\Bbb K$ the reals, complex numbers, quaternions, or octonions, builds a spherical $t$-design by placing a spherical $t$-design on each $\Bbb K$-projective or $\Bbb K$-Hopf fiber associated to the points of a $\lfloor t/2\rfloor$-design on a quotient projective space $\Bbb{KP}^n\neq\Bbb{OP}^2$ or sphere. This generalizes work of K\"{o}nig and Kuperberg, who verified the $\Bbb K=\Bbb C$ case of the projective settings, and of Okuda, who (inspired by independent observation of this construction by Cohn, Conway, Elkies, and Kumar) verified the $\Bbb K=\Bbb C$ case of the generalized Hopf settings.