FINE GRADINGS ON THE EXCEPTIONAL LIE ALGEBRA d4

We describe all the fine group gradings, up to equivalence, on the Lie algebra d4. This problem is equivalent to finding the maximal abelian diagonalizable subgroups of the automorphism group of d4. We prove that there are fourteen by using two different viewpoints. The first approach is computational: we get a full description of the gradings by using a particular implementation of the automorphism group of the Dynkin diagram of d4 and some algebraic groups stuff. The second approach, more qualitative, emphasizes some algebraic aspects, as triality, and it is mostly devoted to gradings involving the outer automorphisms of order three.