I'm following Miranda's book on algebraic curves and Riemann surfaces. In the section where he talks about the automorphisms group of the complex tori he claims the following:
Let $L$ be a lattice in $\mathbb{C}^2$ of rank 2. Let $\gamma$ such that $\gamma L = L$. Then $\gamma$ is a root of unity. Now choose $\ell \in L$ a nonzero element of minimal length. If $\gamma \neq 1, -1$ then $\ell$ and $\gamma \ell$ generate the lattice $L$. It follows that $\gamma^2 \ell \in L$ and $\gamma$ is the root of a polynomial of degree at most 2.
Well, it is clear that $\gamma$ has to be a root of unity because it must map an element of minimal length to another one of the same length, otherwise we couldn't have $\gamma L = L$. It is also clear that $\gamma^n = 1$ for some $n$ because there is a finite amount of elements of minimal length. Now I see that $\ell$ and $\gamma \ell$ must be two nonzero elements of $L$ of minimal length linearly independent over $\mathbb{R}$. But I don't see any reason a priori for them to generate $L$. The rest of affirmations are clear. Any help would be greatly appreaciated.