  The aim of this article is to study (additively) indecomposable algebraic
integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally
positive quadratic forms with coefficients in $\mathcal O_K$. There are given
sufficient conditions for an indecomposable element of a quadratic subfield to
remain indecomposable in the biquadratic number field $K$. Furthermore,
estimates are proven which enable algorithmization of the method of escalation
over $K$. These are used to prove, over two particular biquadratic number
fields $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $\mathbb{Q}(\sqrt{6}, \sqrt{19})$,
a lower bound on the number of variables of a universal quadratic forms,
verifying Kitaoka's conjecture.
