So, in the book "The Sensual (Quadratic) Form" By Conway and Fung, together with the notion of the base for the lattice, there is also a notion of superbase:
"In the same spirit, a strict base is an ordered pair $(e_1, e_2)$ whose integral linear combinations are exactly all the lattice vectors. A lax base is a set $\{\pm e_1, \pm e_2\}$ obtained from a strict base. Finally, a strict superbase is an ordered triple $(e_1, e_2, e_3)$, for which $e_1+e_2+e_3 = 0$ and $(e_1, e_2)$ is a strict base(i.e., with strict vectors), and a lax superbase is a set $\{\pm e_1, \pm e_2, \pm e_3\}$ where $(e_1, e_2, e_3)$ is a strict superbase".
And then there is the following statement:
"On the other hand, each base $\{\pm e_1, \pm e_2\}$ is in just two superbases: $$ \{\pm e_1, \pm e_2, \pm (e_1+e_2)\},\ \ \{\pm e_1, \pm e_2, \pm (e_1-e_2)\}. $$ Note that each of these really is a superbase".
Can you please explain, why each of these form a (lax) superbase? For I understand that if $(e_1,e_2,e_1+e_2)$, then $2e_1 + 2e_2 = 0$, which cannot be the case.
If $(e_1,e_2)$ is a strict base, then so are $(-e_1,e_2)$, $(e_1,-e_2)$ and $(-e_1,-e_2)$. These all induce the same lax base $\{\pm e_1,\pm e_2\}$.
Next, given a strict base $(e_1,e_2)$, there is a unique superbase obtained from it: $(e_1,e_2,-e_1-e_2)$. The superbases related to the other strict bases above are $(-e_1,e_2,e_1-e_2)$, $(e_1,-e_2,-e_1+e_2)$, and $(-e_1,-e_2,e_1+e_2)$.
The superbases $(e_1,e_2,-e_1-e_2)$ and $(-e_1,-e_2,e_1+e_2)$ induce the same lax superbase: $$\{\pm e_1,\pm e_2, \pm(e_1+e_2)\},$$ and $(-e_1,e_2,e_1-e_2)$ and $(e_1,-e_2,-e_1+e_2)$ induce the same lax superbase: $$\{\pm e_1,\pm e_2,\pm(e_1-e_2)\}.$$