Hyperbolicity of algebras with involution over a given extension

Let F be a field and (A,σ) a central simple F-algebra with involution. Let π(t) be a separable polynomial over F. Let F(π)=Ft]/(π(t)). Tignol considered the question whether the algebra A⊗F(π) is hyperbolic. He introduced the algebra H_π, which is universal for this question. Haile and Tignol determined the structure of the algebra H_π and introduced a certain homomorphic image C_π of H_π. In this paper, we give a new characterization of C_π and introduce a new algebra A_π that classifies the commutative algebras with involution that become hyperbolic over F(π). We determine the structure of A_π and use it to examine some examples.