A Coordinatization Theorem for the Jordan algebra of symmetric 2x2 matrices

The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra Hₙ(F) of symmetric nxn matrices over a field F with the same identity element, for n≥ 3. In this paper we extend the Jacobson Coordinatization Theorem for n=2. Specifically, we prove that if J is a unitary Jordan algebra containing the Jordan matrix algebra H₂(F) with the same identity element, then J has a form J=H₂(F)⊗ A₀+k⊗ A₁, where A=A₀+A₁ is a Z₂-graded Jordan algebra with a partial odd Leibniz bracket {,} an k=e₁₂-e₂₁∈ M₂(F) with the multiplication given by (a⊗ b)(c⊗ d)=ac⊗ bd + [a,c]⊗ b,d, the commutator [a,c] is taken in M₂(F).