  We study totally positive definite quadratic forms over the ring of integers
$\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m},
\sqrt{s})$. We restrict our attention to classical forms (i.e., those with all
non-diagonal coefficients in $2\mathcal{O}_K$) and prove that no such forms in
three variables are universal (i.e., represent all totally positive elements of
$\mathcal{O}_K$). This provides further evidence towards Kitaoka's conjecture
that there are only finitely many number fields over which such forms exist.
  One of our main tools are additively indecomposable elements of
$\mathcal{O}_K$; we prove several new results about their properties.
