  We study quadratic forms over totally real number fields by using an
associated ring of quaternions. We examine some properties of residue class
rings of these quaternions and use geometry of numbers to prove that certain
ideals of the ring of quaternions contain elements of a small norm. We prove
that $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal over
$\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ and that $x^2+xy+y^2+z^2+zw+w^2$ represents
all totally positive multiples of certain special elements.
