Macdonald’s Theorem says
that if an identity in three variables x,y,z which is linear in z holds for all special
Jordan algebras, it holds for all Jordan algebras. We show this is equivalent to saying
the universal quadratic envelope 𝒰𝒬ℰ(F(2)) of the free Jordan algebra F(2) on two
generators x,y is canonically isomorphic to the universal compound linear envelope
𝒰𝒬ℰ(F(2)). We generalize Macdonald’s Theorem from the case of linear Jordan
algebras over a field of characteristic ≠2 to quadratic Jordan algebras over an
arbitrary ring of scalars, at the same time improving on the results in the linear
case by presenting 𝒰𝒬ℰ(F(2)) in terms of a finite number of generators and
relations. Similarly we generalize Macdonald’s Theorem with Inverses concerning
identities in x,x−1,y,y−1,z. Finally, we prove Shirshov’s Theorem that F(2) is
special.
