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

Decompositions of preduals of JBW and JBW$^*$ algebras