  The doubling construction is a fast and important way to generate new
solutions to the Hurwitz problem on sums of squares identities from any known
ones. In this short note, we generalize the doubling construction and obtain
from any given admissible triple $[r,s,n]$ a series of new ones
$[r+\rho(2^{m-1}),2^ms,2^mn]$ for all positive integer $m$, where $\rho$ is the
Hurwitz-Radon function.
