Extending involutions on Frobenius algebras

Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C 2 -module for this action, then B is a Frobenius algebra.