Palindromization and construction of Markoff triples

The Markoff equation is the Diophantine equation x2+y2+z2=3xyz. A solution is called a Markoff triple. We give a bijection between the free monoid on two letters and the set of Markoff triples, using the palindromization map of Aldo de Luca. In our construction, special Christoffel words appear, whose lengths are Markoff numbers; we study their standard and palindromic factorizations, and show that they are self-dual.