M Theory Lesson 126

As Ebeling explains, by taking the seven rows of the Hamming circulant and adding a check bit to each row we can write down the seven vectors

$v_1 = \frac{1}{\sqrt{2}} (0,1,1,0,1,0,0,1)$
$v_2 = \frac{1}{\sqrt{2}} (0,0,1,1,0,1,0,1)$
$v_3 = \frac{1}{\sqrt{2}} (0,0,0,1,1,0,1,1)$
$v_4 = \frac{1}{\sqrt{2}} (1,0,0,0,1,1,0,1)$
$v_5 = \frac{1}{\sqrt{2}} (0,1,0,0,0,1,1,1)$
$v_6 = \frac{1}{\sqrt{2}} (1,0,1,0,0,0,1,1)$
$v_7 = \frac{1}{\sqrt{2}} (1,1,0,1,0,0,0,1)$

in $\mathbb{R}^{8}$. These satisfy the rule for a root lattice, $v^{2} = 2$. With the change of variables $e_1 = v_1$, $e_2 = v_2 - v_1$, $e_3 = v_3 - v_2$, $e_4 = v_4 - v_3$, $e_5 = v_5 - v_4$, $e_6 = v_6 - v_5$ and $e_7 = v_7 - v_6$, and the addition of the vector

$e_8 = \frac{1}{\sqrt{2}} (-1,-1,0,0,1,0,-1,0)$

we have a basis for the $E8$ lattice. Since $u^{2} = 2$ for such vectors, it follows that $u \cdot v \in \{ 0, \pm 1, \pm 2 \}$. Hopefully these numbers are familiar.