Classification of hyperbolic planes in algebras of degree three

We classify hyperbolic planes in the space of reduced trace zero elements in a division algebra of degree 3 into three basic types, a description arising from the study of certain algebras associated to these planes. In particular we prove that every division algebra of degree three admits hyperbolic planes of all three types. We also prove a symmetric version of these results for those division algebras that admit an involution of the second kind.