  We show that for all real biquadratic fields not containing $\sqrt{2}$,
$\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7},$ and $\sqrt{13}$, the Pythagoras
number of the ring of algebraic integers is at least $6$. We will also provide
an upper bound on the norm and the minimal (codifferent) trace of additively
indecomposable integers in some families of these fields.
