We study the structure of two-fold self-joinings for a typical transformation of the standard non-atomic Borel space $(X,\mu)$. We prove that the typical automorphism is not simple and semisimple. This is an answer to questions posed by del Junco, Rudolph, Lemanczyk and Mentzen. In order to answer these questions we proved that for a typical transformation there exists an ergodic self-joining concentrated on two fibers, i.e. there exists the simplest non-trivial ergodic self-joining.