A short note on universality of some quadratic forms

In a paper by J. Deutsch [1], a quaternionic proof of the universalityof seven quaternary quadratic forms was given. The proof relieson a construction very similar to that of Hurwitz quaternions, and itsassociated division algorithm. Of course, these results are evident, ifone uses the Conway-Schneeberger Fifteen Theorem [2], as the authoralso mentioned, however it is interesting to give a direct proof for somespecific quadratic forms based on simple argument. It is the purposeof this short note to prove five of the seven quadratic forms mentionedand proven by Deutsch, using the universality of the classical quadraticform associated to the celebrated Lagrange’s Theorem of Four Squaresand Euler’s trick.