Lie maps on alternative rings preserving idempotents

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