  Let $D$ be a positive definite quaternion algebra over a totally real number
field $K$, $F(X,Y)$ a hermitian form in 2N variables over $D$, and $Z$ a right
$D$-vector space which is isotropic with respect to $F$. We prove the existence
of a small-height basis for $Z$ over $D$, such that $F(X,X)$ vanishes at each
of the basis vectors. This constitutes a non-commutative analogue of a theorem
of Vaaler, and presents an extension of the classical theorem of Cassels on
small zeros of rational quadratic forms to the context of quaternion algebras.
