Triality in so

We recall the construction of triality in so(8) given by E. Cartan and we give a matrix definition for this automorphism for the real form so(4,4). Next we compute the induced results on the characteristic classes (Pontryagin and Stiefel-Whitney classes). Paralelly we study the triality automorphism of the singularity D4 (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of G2 singularity.