  Let $G$ be a simple algebraic group over an algebraically closed field of
characteristic $p\ge 0$ and $\mathfrak{g}={\rm Lie}(G)$. We show that the
nilpotent pieces ${\rm LX}(\Delta)$ introduced by Lusztig form a partition of
the nilpotent cone of $\mathfrak{g}$ and hence coincide with the corresponding
Hesselink strata $\mathcal{H}(\Delta)$. Here $\Delta$ runs through the set of
all weighted Dynkin diagrams of $G$. Thanks to the results obtained by Lusztig,
Xue and Voggesberger this boils down to establishing the partition property of
the pieces ${\rm LX}(\Delta)$ for groups of type ${\rm E_7}$ in characteristic
$2$ and for groups of type ${\rm E_8}$ in characteristic $2$ and $3$.
