Double centralizers of unipotent elements in simple algebraic groups of type $E_7$ and $E_8$

This article addresses questions about the double centralizer of unipotent elements u in simple algebraic groups G of type and defined over algebraically closed fields of bad characteristic. We use the method developed in [14] to determine , deduce its dimension and recognize if it is an overgroup for u. The method used requires explicit representatives of the component group of which we produce in all cases. This article extends the results of [14] to all exceptional type groups.