In this paper we extend to
quadratic Jordan algebras certain results due to P. M. Cohn giving conditions under
which a Jordan algebra is special, the most important of ihese being the
Shirshov-Cohn Theorem that a Jordan algebra with two generators and no extreme
radical is always special. We also prove that the free algebra on two generators x, y
modulo polynomial relations p(x) = 0,q(y) = 0 is special, and by taking a particular
p(x) we show that most of the properties of the Peirce decomposition of a Jordan
algebra relative to a supplementary family of orthogonal idempotents follow
immediately from the analogous properties of Peirce decompositions in associative
algebras.
