Composition algebras and cyclic p-algebras in characteristic 3

Given a central simple algebra A of degree 3 over a field of characteristic 3, we prove that there is a unique symmetric composition algebra extending the commutator operation on the trace-zero part modulo scalars. This is analogous to Okubo's construction of symmetric composition algebras in the case of characteristic not 3. We apply the composition algebra tools to obtain a classification of maximal 3-central spaces and maximal Galois hyperplanes of A, and prove a new common slot lemma for such algebras.