  A recursive construction is presented for the projective cubature formulas of
index $p$ on the unit spheres $S(m,K)\subset K^m$ where $K$ is $R$ or $C$, or
$H$. This yields a lot of new upper bounds for the minimal number of nodes
$n=N_K(m,p)$ in such formulas or, equivalently, for the minimal $n$ such that
there exists an isometric embedding $\ell_{2; K}^m \rightarrow \ell_{p; K}^n$.
