  We show that if $B$ is a block of a finite group algebra $kG$ over an
algebraically closed field $k$ of prime characteristic $p$ such that $\HH^1(B)$
is a simple Lie algebra and such that $B$ has a unique isomorphism class of
simple modules, then $B$ is nilpotent with an elementary abelian defect group
$P$ of order at least $3$, and $\HH^1(B)$ is in that case isomorphic to the
Jacobson-Witt algebra $\HH^1(kP)$. In particular, no other simple modular Lie
algebras arise as $\HH^1(B)$ of a block $B$ with a single isomorphism class of
simple modules.
