  Let $H$ be a linear algebraic group over an algebraically closed field of
characteristic $p>0$. We prove that any "exponential map" for $H$ induces a
bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent
elements in $Lie(H)$ and the variety of height $r$ infinitesimal one-parameter
subgroups of $H$. In particular, we show that for a connected reductive group
$G$ in pretty good characteristic, there is a canonical exponential map for $G$
and hence a canonical bijection between the aforementioned varieties, answering
in this case questions raised both implicitly and explicitly by Suslin,
Friedlander, and Bendel.
