A note on the structure of complete alternative local algebras

Let (A, m) be an alternative algebra with maximal ideal m which is complete and separated for the m-adic topology. Assuming that A/m := k is a perfect field of positive characteristic and that the associated graded algebra is a k-algebra, we show that the reduction map W(k) → k from the Witt ring W(k) lifts canonically to a morphism W(k) → A thereby giving A the structure of a unital W(k)-algebra.