Double centralizers of unipotent elements in simple algebraic groups of type G2, F4 and E6

Let u be a unipotent element of a simple algebraic group G over a field k of characteristic p. We develop a method for computing the connected component of Z(CG(u)) in the cases where p is positive and both it and the rank of G are small enough. The method is then carried out for G of type G2, F4 and E6 in bad characteristic.