  Let $\Re$ and $\Re'$ unital $2$,$3$-torsion free alternative rings and
$\varphi: \Re \rightarrow \Re'$ be a surjective Lie multiplicative map that
preserves idempotents. Assume that $\Re$ has a nontrivial idempotents. Under
certain assumptions on $\Re$, we prove that $\varphi$ is of the form $\psi +
\tau$, where $\psi$ is either an isomorphism or the negative of an
anti-isomorphism of $\Re$ onto $\Re'$ and $\tau$ is an additive mapping of
$\Re$ into the centre of $\Re'$ which maps commutators into zero.
