  Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a
finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha)$, $G(3)$ or
$F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic
group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$.
Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer
$\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre
$\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled
Dynkin diagram for $e$. We prove theorems relating the dimension of
$\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and the labelled Dynkin
diagram.
