  Various reconstructions of finite-dimensional quantum mechanics result in a
formally real Jordan algebra A and a last step remains to conclude that A is
the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is
shown that this can be achieved by postulating that there is a locally
tomographic model for a composite system consisting of two copies of the same
system. Local tomography is a feature of classical probability theory and
quantum mechanics; it means that state tomography for a multipartite system can
be performed by simultaneous measurements in all subsystems. The quantum
logical definition of local tomography is sufficient, but it is less restictive
than the prevalent definition in the literature and involves some subtleties
concerning the so-called spin factors.
