General preservers of quasi-commutativity on self-adjoint operators

Let H be a separable Hilbert space and B sa ( H ) the set of all bounded linear self-adjoint operators. We say that A , B ∈ B sa ( H ) quasi-commute if there exists a nonzero ξ ∈ C such that A B = ξ B A . Bijective maps on B sa ( H ) which preserve quasi-commutativity in both directions are classified.