  Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or
$\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let
$\mathfrak{g}=\mathrm{Lie}(\bar{G})=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$
and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as a
superalgebra concentrated in even degree. Suppose $e\in\mathfrak{g}_{\bar{0}}$
is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in
$\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$. In particular,
we give bases for $\mathfrak{g}^{e}$, $\mathfrak{z}(\mathfrak{g}^{e})$ and
$\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$. We also determine the
labelled Dynkin diagram $\varDelta$ with respect to $e$ and subsequently
describe the relation between
$\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and $\varDelta$.
