Lie maps on alternative rings preserving idempotents

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.