Constructions of lattices via biquadratic and triquadratic fields

The lattices $D_4$ and $E_8$ are known to be the densest lattices in dimensions 4 and 8, respectively. In this paper, we employ tools from algebraic number theory to prove that the $D_4$-lattice arises from an infinite family of totally imaginary biquadratic fields. Furthermore, we extend this construction to show that both the $D_8$ and $E_8$ can be realized via triquadratic fields derived from this family.