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 $Z(C_G(u))$ 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 $G_2$, $F_4$ and $E_6$ in bad characteristic.