  We prove that the predual of any JBW$^*$-algebra is a complex $1$-Plichko
space and the predual of any JBW-algebra is a real $1$-Plichko space. I.e., any
such space has a countably $1$-norming Markushevich basis, or, equivalently, a
commutative $1$-projectional skeleton. This extends recent results of the
authors who proved the same for preduals of von Neumann algebras and their
self-adjoint parts. However, the more general setting of Jordan algebras turned
to be much more complicated. We use in the proof a set-theoretical method of
elementary submodels. As a byproduct we obtain a result on amalgamation of
projectional skeletons.
