D₄ root lattice - Knowlpedia

The $D_4$ root lattice, with its standard Euclidean normalization, is

D_4=\{x\in\mathbb Z^4:x_1+x_2+x_3+x_4\in2\mathbb Z\}.

It is the integer span of the roots $\pm e_i\pm e_j$ for $1\le i<j\le4$ , using the standard basis of $\mathbb R^4$ . These are its twenty-four shortest nonzero vectors, all of squared length $2$ .

Hurwitz coordinates and scale

The Euclidean dual is

D_4^*=\{y:\langle y,D_4\rangle\subseteq\mathbb Z\} =\mathbb Z^4\cup\left(\mathbb Z+\tfrac12\right)^4.

Under the coordinates $a+bi+cj+dk\leftrightarrow(a,b,c,d)$ , this is the additive lattice of Hurwitz quaternions with quadratic norm $a^2+b^2+c^2+d^2$ .

The map

(x_1,x_2,x_3,x_4)\longmapsto \tfrac12(x_1+x_2,x_1-x_2,x_3+x_4,x_3-x_4)

sends $D_4$ onto $D_4^*$ and divides squared lengths by $2$ . Thus the Hurwitz lattice is similar to $D_4$ with length scale $1/\sqrt2$ , not length scale $1/2$ . Its twenty-four shortest vectors are the Hurwitz units.

Direct verification

Pairing with $e_i-e_j$ shows that dual coordinates have the same fractional part; pairing with $e_i+e_j$ shows that it is $0$ or $1/2$ . In the displayed map, the even-sum condition makes all four output coordinates integral or all half-integral, and the inverse sends either case to an even-sum integer vector. This proves both the dual description and the scale statement.