  The aim of this paper is to give another proof to a result on the image of a
homogeneous quadratic map which is positive with respect to a homogeneous cone,
given by Graczyk and Ishi in 2014. The new proof depends on a purely algebraic
method, whereas the original depends on analytic arguments. This enables us to
give explicit formulas of the basic relative invariants of homogeneous cones,
obtained by the previous paper [J. Lie Theory 24 (2014), 1013--1032] without
analytic arguments.
