Preduals of JBW${}^{\ast }$-triples are 1-Plichko spaces

We prove that the predual, $M_{\ast }$, of a JBW${}^{\ast }$-triple $M$ is a 1-Plichko space (i.e. it admits a countably 1-norming Markushevich basis or, equivalently, it has a commutative 1-projectional skeleton), and obtain a natural description of the $\mathrm{\Sigma }$-subspace of $M$. This generalizes and improves similar results for von Neumann algebras and JBW${}^{\ast }$-algebras. Consequently, dual spaces of JB${}^{\ast }$-triples also are 1-Plichko spaces. We also show that $M_{\ast }$ is weakly Lindel"{o}f determined if and only if $M$ is $\sigma$-finite if and only if $M_{\ast }$ is weakly compactly generated. Moreover, contrary to the proof for JBW${}^{\ast }$-algebras, our proof dispenses with the use of elementary submodels theory.