Abstract
An algebra of rank three is a commutative, finite dimensional algebra that may be defined by the property that every element generates a subalgebra of dimension not greater than two. In this article we discuss several classes of such algebras, including two classes related to central simple Jordan algebras, and derive some general results which indicate that, with the exception of one pathological class related to nilpotent algebras, every rank three algebra can be constructed either from a quadratic and alternative algebra or from a representation of a Clifford algebra. Among other results, semisimple and simple rank three algebras are characterized, and the radical of an arbitrary rank three algebra is determined.
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