The lattices $D_n$ and $A_n$ are integral. There is a result by krüsskemper stating that any integral lattice can be constructed as an ideal lattice in some real number field. The proof is constructive and based on the previous work of O. Taussky, so in principle these lattices can be explicitly obtained as sublattices of some 'algebraic' lattices in any dimension. Other methods were applied to obtain such lattices over number fields, see for instance Theorem 3.20 in Damir-Mantilla Soler where the lattice consisting of zero trace elements in a tame number field of either conductor or degree prime is a scaled $A_n$. For the $D_n$ lattice see for example Jorge.

The lattices $D_n$ and $A_n$ are integral. There is a result by Krüskemper (Algebraic construction of bilinear forms) stating that any integral lattice can be constructed as an ideal lattice in some real number field. The proof is constructive and based on the previous work of O. Taussky (On the similarity transformation between an integral matrix with irreducible characteristic polynomial and its transpose), so in principle these lattices can be explicitly obtained as sublattices of some 'algebraic' lattices in any dimension. Other methods were applied to obtain such lattices over number fields, see for instance Theorem 3.20 in Damir–Mantilla-Soler - Bases of minimal vectors in tame lattices where the lattice consisting of zero trace elements in a tame number field of either conductor or degree prime is a scaled $A_n$. For the $D_n$ lattice see for example de Araujo and Jorge - Construction of full diversity $D_n$-lattices for all $n$.