Sturmian morphisms, the braid group B4, Christoffel words and bases of F2

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F2) of automorphisms of the rank two free group F2 and show that it can be realized as a monoid in the group B4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F2 lifting any given basis of the free abelian group Z2. We further give an algorithm allowing to decide whether two elements of F2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.