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The aim of this paper is to give another proof to a result on the image of ahomogeneous 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 algebraicmethod, whereas the original depends on analytic arguments. This enables us togive explicit formulas of the basic relative invariants of homogeneous cones,obtained by the previous paper [J. Lie Theory 24 (2014), 1013--1032] withoutanalytic arguments.

Algebraic proof of explicit formulas of basic relative invariants of homogeneous cones