Designs related through projective and Hopf maps

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